Fractions
In the sexagesimal system, any fraction in which the denominator is a regular number (having only 2, 3, and 5 in its prime factorization) may be expressed exactly. The table below shows the sexagesimal representation of all fractions of this type in which the denominator is less than 60. The sexagesimal values in this table may be interpreted as giving the number of minutes and seconds in a given fraction of an hour; for instance, 1/9 of an hour is 6 minutes and 40 seconds. However, the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.
| Fraction: | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | 1/8 | 1/9 | 1/10 |
|---|---|---|---|---|---|---|---|---|
| Sexagesimal: | 30 | 20 | 15 | 12 | 10 | 7:30 | 6:40 | 6 |
| Fraction: | 1/12 | 1/15 | 1/16 | 1/18 | 1/20 | 1/24 | 1/25 | 1/27 |
| Sexagesimal: | 5 | 4 | 3:45 | 3:20 | 3 | 2:30 | 2:24 | 2:13:20 |
| Fraction: | 1/30 | 1/32 | 1/36 | 1/40 | 1/45 | 1/48 | 1/50 | 1/54 |
| Sexagesimal: | 2 | 1:52:30 | 1:40 | 1:30 | 1:20 | 1:15 | 1:12 | 1:6:40 |
However numbers that are not regular form more complicated repeating fractions. For example:
- 1/7 = 0:8:34:17:8:34:17 ... (with the sequence of sexagesimal digits 8:34:17 repeating infinitely often).
The fact in arithmetic that the two numbers that are adjacent to 60, namely 59 and 61, are both prime numbers implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as their denominators, and that other non-regular primes have fractions that repeat with a longer period.
Read more about this topic: Sexagesimal