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# Math in Ancient Mesopotamia

When the Sumerians first came up with their numerals, they did not have a specific symbol for zero. If they needed to inscribe, say, 506 on a clay tablet, they would simply put a blank space between the symbols of 5 () and 6 ().

This way of denoting zero could be quite confusing and problematic. Neither the Sumerians nor other people in Mesopotamia (most notably, the Babylonians) were able to come up with a solution at the time. This issue would remain unsolved until around 500 A.D. when the Indians developed the Arabic numerals that we are still using today.

In the Mesopotamian civilization, we find, differently from other ancient civilizations, a positional system of basis 60. One can argue that the choice of such basis occurred by chance, or as the merge of a system of basis 10 with one of basis 6, or by any other natural” process.

A much more plausible hypothesis is that it had its origins from practical measurements, since the number 60, with its many divisors, is a comfortable choice in commercial activities. In this hypothesis, even though motivated by a utilitarian need, the choice of the basis took into account a general and intrinsic feature of numbers: the quantity of their divisors.

The ancient Mesopotamians did not have a money economy, so they developed a standardized system of weights to carry out their many commercial transactions. The original medium of exchange was barley. The smallest unit of weight was called a barleycorn, the approximate weight of one grain of barley.

The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25′ 30″, i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number, for more details regarding this notation see our article on Babylonian numerals. As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation.

Mesopotarnian mathematicians were the most skllled algebraists of the ancient world. They were able to solve any quadratic equation and many cubic equations. It is possible that their methods diffused to India and from there back to the Arab world, from which algebra reached the West. It was once fashionable to say that the Mesopotamians were good in algebra but weak in geometry. Later discoveries have forced a revision of this belief, since it is clear that the Mesopotamians were the earliest people to know the Pythagorean theorem (Pythagoras, who is known to have traveled in the East, may have learned his famous theorem there).