Regular Number - Babylonian Mathematics

Babylonian Mathematics

In the Babylonian sexagesimal notation, the reciprocal of a regular number has a finite representation, thus being easy to divide by. Specifically, if n divides 60k, then the sexagesimal representation of 1/n is just that for 60k/n, shifted by some number of places.

For instance, suppose we wish to divide by the regular number 54 = 2133. 54 is a divisor of 603, and 603/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/602 + 40/603, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/603, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places.

The Babylonians used tables of reciprocals of regular numbers, some of which still survive (Sachs, 1947). These tables existed relatively unchanged throughout Babylonian times.

Although the primary reason for preferring regular numbers to other numbers involves the finiteness of their reciprocals, some Babylonian calculations other than reciprocals also involved regular numbers. For instance, tables of regular squares have been found and the broken cuneiform tablet Plimpton 322 has been interpreted by Neugebauer as listing Pythagorean triples generated by p, q both regular and less than 60.