Babylonian Mathematics

The Babylonians, who lived in Mesopotamia from about 2000 B.C. until the last few centuries B.C., developed a substantial body of mathematical knowledge, well before most other civilizations. Their writings have been passed down to us on hundreds of clay tablets, on which they wrote in cuneiform (wedge-shaped) symbols. Most of the Babylonian mathematics that has come down to us on these tablets is from the period 1800-1600 B.C.; it is not clear whether mathematical understanding progressed significantly after that period.

Many of the mathematical tablets of the Babylonians consist of arithmetical tables. These tables, which are easier to translate than more complicated texts, proved invaluable to scholars who set out to decipher Babylonian writings. The tables made it clear that the Babylonians used a base-60 or sexagesimal system in place of the base-10 system in use today. Part of that system has been preserved even to the present in the way we measure time: sixty minutes to each hour, sixty seconds to each minute. In some ways, the base-60 system is much more cumbersome than the base-10 system. Instead of ten numerals (0, 1,..., 9), sixty are needed. This made learning the number system such a challenge for the Babylonians that it was generally reserved for the priesthood. On the other hand, 60 has a distinct advantage over 10, because it has many more divisors: 10 is only divisible by 2 and 5, while 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. This means that many fractions that have infinite decimal representations have finite sexagesimal representations. For example, the fraction 1/3 has the infinite decimal expansion 0.33333..., since 1/3=3/10+3/100+3/1000+.... But 60, unlike 10, is divisible by 3, and so 1/3=20/60, and 1/3 has a finite sexagesimal representation.

The Babylonians drew up tables of squares, square roots, cube roots, reciprocals, and some powers. They were adept at linear and quadratic equations and knew many geometric constructions. They had tables of logarithms and an approximation for the natural logarithm "e." They estimated pi and were aware of the Pythagorean theorem, as is shown in the following tablet inscription:

"4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth."

This document shows that although the Babylonians had a good understanding of how to apply the theorem, they tended to think algorithmically; that is, in terms of a sequence of steps that could always be followed to arrive at the answer to a problem. They did not attempt any formal proof of the statement; that would not come until much later, when Pythagoras offered a proof in the 6th century B.C.