Mathematics

The peoples of ancient Mesopotamia— particularly the sumerians and the Baby-lonians—made important contributions to the early development of mathematics, and some of their mathematical ideas and systems are still used today. The early Sumerians had a simple decimal system, one based on the number 10, that they probably inherited from earlier peoples. That system used small clay tokens to indicate various numbers. One token stood for 1 sheep, 1 measure of grain, or, in the more general sense, the number 1. Another token stood for 10 sheep or the number 10. In this system, a number such as 23 was denoted by two number-10 tokens and three number-1 tokens.

Over time the Sumerians began writing numerical symbols on clay tablets, each symbol indicating a certain token and the number it represented. These symbols became very complex and are still not completely understood. By the last centuries of the third millennium b. c., however, the scribes had simplified the system, narrowing it down to just two symbols, a vertical wedge and a corner wedge. These had different meanings depending on their placement. In one configuration, they indicated 60 (6 x 10); in another, they stood for 3,600 (60 x 60), and so forth. All the multiples used either 6 or 10, which made it a rudimentary sexagesimal system (one based on the number 60). Everyday calculations, for commerce or construction, became somewhat less cumbersome when the Akkadians introduced the abacus in the late third millennium B. C.

Later, the Babylonians inherited and continued to use the Sumerian decimal system for counting, but only for numbers from 1 to 59. For higher numbers, the Babylonians expanded the Sumerian sexagesimal system into a more sophisticated version. Because the new system combined elements of decimal and sexagesimal schemes, it was rather clumsy in comparison to the simpler, strictly decimal version used in most of the world today. In Babylonia, for example, the number 3,832 was expressed as 1,3,52. Reading from right to left, the second number occupied an order of magnitude higher than the first, and the third an order higher than the second. Thus, the user understood that the 1 stood for 1 x 602 (or 3,600), the 3 for 3 x 60 (or 180), and the 52 for 52 single decimal units. Adding the three numbers together therefore rendered 3,600 + 180 + 52 = 3,832. This shows that the Babylonians understood and used square roots. They also employed cube roots (for instance, 603 = 60 x 60 x 60 = 216,000). The following is an actual surviving math problem from ancient Babylonia:

Problem: If somebody asks you thus:

As much as the side of the square which I made I dug deep, and I extracted one musaru [603] and a half of volume of earth. My base (ground)

I made a square. How deep did I go? Solution: You, in your procedure, operate with 12. Take the reciprocal of 12 and multiply by 1,30,0,0 which is your volume. 7,30,0 you will see [i. e., will be the answer]. What is the cube root of 7,30,0? 30 is the cube root. Multiply 30 by 1, and 30 you [will] see. Multiply 30 by another 1, and 30 you [will] see. Multiply 30 by twelve, and 6,0 [360] you [will] see.

30 is the side of your square, and 6,0 [360] is your depth.

Some remnants of this sexagesimal system have survived. For instance, people still divide a circle into 360 degrees and count 60 seconds to a minute and 60 minutes to an hour in timekeeping, navigation, and astronomy.

The Babylonians also employed mathematical tables of various kinds to make some calculations easier. They had multiplication tables, tables of square roots and cube roots, and tables that listed monetary conversions, including the equivalents of the values of various goods to the value of given weights of silver. The Babylonians also developed a simple form of geometry for determining area and volume. This proved useful in constructing ziggurats and other large-scale buildings. Babylonian geometry eventually influenced early

Greek thinkers, who created a more sophisticated version by introducing provable theorems.

See Also: astrology and astronomy; weights and measures; writing materials