Michigan web site - 180 ARITHMETIC 4.1 During the twentieth century, historians
180 ARITHMETIC 4.1 During the twentieth century, historians of mathematics have made exten- sive studies of early cuneiform tablets found by archeologists in the Middle East. These studies show that the Babylonian people actually had two distinct systems of number representation: Numbers used in everyday business transactions were written in a notation based on grouping by tens, hundreds, etc.; this notation was inherited from earlier Mesopotamian civilizations, and large numbers were seldom required. When more difficult mathematical problems were considered, however, Babylonian mathematicians made extensive use of a sexagesimal (radix sixty) positional notation that was highly developed at least as early as 1750 B.C. This notation was unique in that it was actually a Aoatingpoint form of representation with exponents omitted; the proper scale factor or power of sixty was to be sup- plied by the context, so that, for example, the numbers 2, 120, 7200, and & were all written in an identical manner. The notation was especially convenient for multiplication and division, using auxiliary tables, since radix-point alignment had no effect on the answer. As examples of this Babylonian notation, consider the following excerpts from early tables: The square of 30 is 15 (which may also be read, The square of 4 is a ); the reciprocal of 81 = (1 21)ao is (44 26 4O)eo; and the square of the latter is (32 55 18 31 6 4O)so. The Babylonians had a sym- bol for zero, but because of their floating point philosophy, it was used only within numbers, not at the right end to denote a scale factor. For the interesting story of early Babylonian mathematics, see 0. Neugebauer, The Exact Sciences in Antiquity (Princeton, N. J.: Princeton University Press, 1952), and B. L. van der Waerden, Science Awakening, tr. by A. Dresden (Groningen: P. Noordhoff, 1954); see also D. E. Knuth, CACM 15 (1972), 671-677; 19 (1976), 108. Fixed point positional notation was apparently first conceived by the Maya Indians in central America 2000 years ago; their radix-20 system was highly developed, especially in connection with astronomical records and calendar dates. But the Spanish conquerors destroyed nearly all of the Maya books on history and science, so we have comparatively little knowledge about how sophisticated the native Americans had become at arithmetic; special-purpose multiplication tables have been found, but no examples of division are known [cf. J. Eric S. Thompson, Contributions to Amer. Anthropology and History 7 (Carnegie Inst. of Washington, 1942), 37-621. Several centuries before Christ, the Greek people employed an early form of the abacus to do their arithmetical calculations, using sand and/or pebbles on a board that had rows or columns corresponding in a natural way to our decimal system. It is perhaps surprising to us that the same positional notation was never adapted to written forms of numbers, since we are so accustomed to reckoning with the decimal system using pencil and paper; but the greater ease of calculating by abacus (since handwriting was not a common skill, and since abacus calculation makes it unnecessary to memorize addition and multiplication tables) probably made the Greeks feel it would be silly even to suggest that computing could be done better on scratch paper. At the same time Greek astronomers did make use of a sexagesimal positional notation for fractions, which they had learned from the Babylonians.