Sexagesimal Numeration
Sexagesimal numeration is a numeral system in which all derived units are based on the number 60 and the powers of 60. The word sexagesimal is derived from the Latin word sexagesimus (sixty). Between 4000-3000 b.c. the Sumerians developed a sexagesimal number system that had an additive decimal (base-10) sequence of "1 to 10, 10 to 60, 60 to 600, 600 to 3600, and so forth". For example, the Sumerian number of 100 was expressed as "1, 40"; that is, one unit of 60 plus 40. The year 2000 would be expressed in this system as three units of 600, plus three units of 60, plus 10 plus 10. Ptolemy and other Alexandrian astronomers also used sexagesimal fractions that are believed to have been one way that led to the persistent use of the base 60 for measures of time and angles. This is in spite of the nearly worldwide adoption of base-10 numeration for most other measurement needs (currency, the metric system, etc.). For instance, 60 seconds equals one minute and 60 arc-minutes equals one arc-degree. Between 3000-2000 b.c. the Babylonians developed a place-value system (from the older Sumerian system) with a base of 60 that contained an incomplete sexagesimal positional notation. The reason why such a system was developed is unclear, but mathematical common sense indicates that one possibility may have been that it contained many divisors (2, 3, 4, and 5, and some multiples) in order to facilitate division. With such a large base it was awkward to have unrelated names for the digits 0, 1, ... 59, so a simple grouping system from base 10 was used for these numbers. It used only two symbols ("a vertical wedge derived from a small cone" for 1 and "a corner wedge derived from a small circle" for 10) instead of the 60 distinct symbols that a base-60 system would normally use. Therefore, for the number 33 the symbol grouping "three corner wedges and three vertical wedges" was used. For larger numbers the vertical wedge was also used as powers of 60 (such as 1 (600, 60 (601), 3600 (602), and so forth). Thus, for the number 90 the grouping of symbols "one vertical wedge and three corner wedges" was used to denote (60 + 10 + 10 + 10).
As a result, the system suffered from ambiguities in representing values that could be resolved only by analyzing the context. In addition, the Babylonians had long used empty spaces to separate one sexagesimal order from the next in written numbers, but special symbols were sometimes also employed for this purpose. Similar symbols were eventually used to indicate the absence of certain orders. Astronomers may have also used the symbols to indicate fractions. The complete development of the system was, therefore, delayed by the lack of a consistent use of these symbols and for a symbol for empty places, i.e. the zero. The Babylonians appear to have developed a placeholder symbol that functioned as a zero by the third century b.c., but its precise meaning and use is still uncertain. The world of mathematics and astronomy owes much to the Babylonians for their development of the sexagesimal system in order to calculate time (the 12 double-hours in a day, minutes, and seconds) and angles (degrees, arc-minutes, and arc-seconds) that is still in practical use today. In fact, the Babylonian year that was composed of 360 days (of 12 months with 30 days each) most likely created the 360 degrees of a circle where each degree of the circle is further divided into 60 arc-minutes, and each arc-minute into 60 arc-seconds.
The sexagesimal number system was familiar to both the Indian and Mesopotamian cultures (where ancient Babylonia was located within Mesopotamia). In these societies it is likely that the sexagesimal system was based on the observation of the planets, specifically Jupiter and Saturn. Every 60 years, Jupiter and Saturn return to the same relative place in the sky. Jupiter takes 12 years to cross all the constellations in the night sky, from which the zodiac signs have been derived. Saturn takes an average 30 years to cross the zodiac. So Jupiter makes 5 complete crossings of the zodiac, while Saturn makes 2 transits in the same sixty-year period. During the sixty-year cycle, the four numbers involved in the revolutions of the planets (2 and 5) and in the number of years for each revolution (12 and 30) are divisible into 60, allowing for a sensible use of the sexagesimal number system.
Today, the sexagesimal number system is an archaic number system. It has been replaced by numbering systems that are much easier to use. The prime example of which is the dominant decimal number system (base 10); and other systems of use in special applications, such as the binary (base 2) and hexadecimal (base 16) number systems that are especially needed for the operation of digital computers.
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