It was used to write both whole numbers and certain fractions (the equivalents of our decimal fractions) and was without doubt the most efficient way of writing numbers in antiquity.
It is a place-value system, like our decimal system, but of base 60 rather than 10. The backbone of Babylonian mathematics is the sexagesimal number system. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia. In respect of content there is scarcely any difference between the two groups of texts. In respect of time they fall in two distinct groups: one Old Babylonian from the centuries about 1600 B.C., the other mainly Seleucid from the last three or four centuries B.C. – drew their essential ingredients from several branches of learning and literature, chief among them mathematics and, for observations, the astronomical diaries, closely linked to the celestial omen texts.īabylonian mathematical texts are plentiful and well edited. The anonymous creators of Babylonian theoretical astronomy – probably of the fourth or fifth century B.C. Indeed, in the course of the last few decades it has become increasingly clear that all Western efforts in the exact sciences are descendants in direct line from the work of the Late Babylonian astronomers. In sexagesimal 147 = 2,27 and squaring gives the number 21609 = 6,0,9.In no domain has the influence of ancient Mesopotamia on Western civilization been more profound and decisive than in theoretical astronomy and, principally through it, mathematics. Here is an example from a cuneiform tablet (actually AO 17264 in the Louvre collection in Paris) in which the calculation to square 147 is carried out. Returning to empty places in the middle of numbers we can look at actual examples where this happens. Although not a very serious comment, perhaps it is worth remarking that if we assume that all our decimal digits are equally likely in a number then there is a one in ten chance of an empty place while for the Babylonians with their sexagesimal system there was a one in sixty chance.
Perhaps we should mention here that later Babylonian civilisations did invent a symbol to indicate an empty place so the lack of a zero could not have been totally satisfactory to them.Īn empty place in the middle of a number likewise gave them problems. How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem – there is little doubt that they had the skills to come up with a solution had the system been unworkable. The context made it clear, and in fact despite this appearing very unsatisfactory, it could not have been found so by the Babylonians. The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help. In the number 1,1 there is a space between them.Ī much more serious problem was the fact that there was no zero to put into an empty position. In the symbol for 2 the two characters representing the unit touch each other and become a single symbol. However, this was not really a problem since the spacing of the characters allowed one to tell the difference. Since two is represented by two characters each representing one unit, and 61 is represented by the one character for a unit in the first place and a second identical character for a unit in the second place then the Babylonian sexagesimal numbers 1,1 and 2 have essentially the same representation. Now there is a potential problem with the system. This is because the 59 numbers, which go into one of the places of the system, were built from a 'unit' symbol and a 'ten' symbol. Now although the Babylonian system was a positional base 60 system, it had some vestiges of a base 10 system within it. However, rather than have to learn 10 symbols as we do to use our decimal numbers, the Babylonians only had to learn two symbols to produce their base 60 positional system. Now of course this comment is based on knowledge of our own decimal system which is a positional system with nine special symbols and a zero symbol to denote an empty place. Often when told that the Babylonian number system was base 60 people's first reaction is: what a lot of special number symbols they must have had to learn.
Here are the 59 symbols built from these two symbols
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