Why is the discoveries of the Sumerian civilization important to modern mathematics?

Papyrus, parchment, paper ... videotape, DVDs, Blu-ray discs — long after all these materials have crumbled to dust, the first recording medium of all, the cuneiform clay tablet of ancient Mesopotamia, may still endure.

Thirteen of the tablets are on display until Dec. 17 at the Institute for the Study of the Ancient World, part of New York University. Many are the exercises of students learning to be scribes. Their plight was not to be envied. They were mastering mathematics based on texts in Sumerian, a language that even at the time was long since dead. The students spoke Akkadian, a Semitic language unrelated to Sumerian. But both languages were written in cuneiform, meaning wedge-shaped, after the shape of the marks made by punching a reed into clay.

Sumerian math was a sexagesimal system, meaning it was based on the number 60. The system “is striking for its originality and simplicity,” the mathematician Duncan J. Melville of St. Lawrence University, in Canton, N.Y., said at a symposium observing the opening of the exhibition.

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A 59 x 59 multiplication table might not seem simple, and indeed is far too large to memorize, so tablets were needed to provide essential look-up tables. But cuneiform numbers are simple to write because each is a combination of only two symbols, those for 1 and 10.

Why the Sumerians picked 60 as the base of their numbering system is not known for sure. The idea seems to have developed from an earlier, more complex system known from 3200 B.C. in which the positions in a number alternated between 6 and 10 as bases. For a system that might seem even more deranged, if it weren’t so familiar, consider this way of measuring length with four entirely different bases: 12 little units, called inches, make a foot, 3 feet make a yard, and 1,760 yards make a mile.

Over a thousand years, the Sumerian alternating-base method was simplified into the sexagesimal system, with the same symbol standing for 1 or 60 or 3,600, depending on its place in the number, Dr. Melville said, just as 1 in the decimal system denotes 1, 10 or 100, depending on its place.

The system was later adopted by Babylonian astronomers and through them is embedded in today’s measurement of time: the “1:12:33” on a computer clock means 1 (x 60-squared) second + 12 (x 60) seconds + 33 seconds.

The considerable mathematical knowledge of the Babylonians was uncovered by the Austrian mathematician Otto E. Neugebauer, who died in 1990. Scholars since then have turned to the task of understanding how the knowledge was used. The items in the exhibition are drawn from the archaeological collections of Columbia, Yale and the University of Pennsylvania.

They include two celebrated tablets, known as YBC 7289 and Plimpton 322, that have played central roles in the reconstruction of Babylonian math. YBC 7289 is a small clay disc containing a rough sketch of a square and its diagonals. Across one of the diagonals is scrawled 1,24,51,10 — a sexagesimal number that corresponds to the decimal number 1.41421296. Yes, you recognized it at once — the square root of 2. In fact it’s an approximation, a very good one, to the true value, 1.41421356.

Below is its reciprocal, the answer to the problem, that of calculating the diagonal of a square whose sides are 0.5 units. This bears on the issue of whether the Babylonians had discovered Pythagoras’s theorem some 1,300 years before Pythagoras did. No tablet bears the well-known algebraic equation, that the squares of the two smaller sides of a right-angled triangle equal the square of the hypotenuse. But Plimpton 322 contains columns of numbers that seem to have been used in calculating Pythagorean triples, sets of numbers that correspond to the sides and hypotenuse of a right triangle, like 3, 4 and 5.

Plimpton 322 is thought to have been written in Larsa, just north of Ur, some 60 years before the city was captured by Hammurabi the lawgiver in 1762 B.C.

Other tablets bear lists of practical problems, like calculating the width of a canal, given information about its other dimensions, the cost of digging it and a worker’s daily wage.

With some tablets the answers are stated without any explanation, giving the impression that they were for show, a possession designed to make the owner seem an academic.