mercoledì 30 luglio 2014

Paleo-Babylonian Love for Mathematics

by Siziliano

Translation in English, for the original text see link, or the end of this article:
Babylonians were outstanding astronomers. Mathematics was considered to be a useful tool-but it might also have been a toy for fun. This might be concluded by fragments of two clay tablets from the asset of the British Museum (BM). Mathieu Ossendrijver (Professor for history of science in the antique world at Humboldt University, Berlin) presents his studies on playing with numbers from the time period of 450 – 200 BC in the actual issue of “J. Cuneiform Studies”. (Mathieu Ossendrijver, The Powers of 9 and Related Mathematical Tables from Babylon,  Journal of Cuneiform Studies 66 (2014), 149-165, ndr)

How come?

Babylonians did not use the decimal system but instead their calculations were based on the sexagesimal system (number 60). Unusual? Just have a look at your wristwatch, consider a circle (360°) or count the months.

Why dividing endless numbers?

Both tables start with an extraordinarily large number which in the following lines is divided by its factors as long as finally the number 1 is reached. Already the first tower of numbers is breath-taking. The first number refers (in our system) to 9 over 46, 946 (9*9*9* … 46-times multiplied)!! Even more impressing is the starting number of the second table: It starts with something that we would describe as 9 over 11 multiplied with 12 over 39 (911 * 1239)!!!! A number with 30 digits! This is the largest number ever noted in cuneiform writing.
Surely, not to explain why a Babylonian sits down and divides and divides such numbers. Ossendrijver: ‘It is improbable that this duty has any relevance for calculations in administration or astronomy, for number in these disciplines were seldom longer than seven digits’. Thus, he concludes it might have been a mathematical exercise. It might also be that ‘this was a mathematical proof that the initial huge number had been calculated correctly. Alternatively, the Babylonian mathematicians might have been on search for rules in number-theory.
Now comes the fun part: At least the second clay tablet does not hold an original calculation, but is a copy of another unknown original! It carries two tiny errors! Would these be part of the original calculation, the error would have consecutively be found in each of the following lines of calculations. What wonder, the formerly made error disappears in one of the following lines (corrected!), as if nothing had happened.
So, who was it? There is some advice: The text was written ‘In the name of Bel and Beltiya [might it be successful]’ Bel = Marduk, Beltiya, his wife. Both were adored in the Esagil temple that was taken as the center of the world by the Babylonians, pointing to this same location for the author as being also in duty in the temple or nearby.
Original location of findings cannot be determined. Both shards are in the Babylon-collection of BM, most of these were excavated by local inhabitants and British ‘adventurers’ (as the Professor identifies them) between 1876 and 1881 – in a non-scientific manner. Most astronomic and mathematical tablets were found in private houses of theoreticians and scientists directly south of the temple.

Late-Babylonian (LB) mathematics (450-100 BC), represented by some sixty cuneiform tablets from Babylon and Uruk, is incompletely known compared to its abundantly preserved, well-studied Old-Babylonian (OB) predecessor (1800-1600 BC).
With the present paper, sixteen fragments from Babylon, probably belonging to 13 different tablets,
are added to this corpus. Two remarkable tablets represent a hitherto unknown class of very large factorization tables that can be adequately described as Babylonian examples of number crunching (Section I). Most other fragments belong to tables with reciprocals (II) and squares (III). Finally, two fragments contain multiplications of one kind or another (IV)[..]