## Mathematical knowledge of the ancient world

Because baked clay tablets with cuneiform symbols impressed are easily preserved, especially in a dry climate, much is known about Mesopotamian mathematics. Some historians think that it is likely that a great deal of the mathematical knowledge of the ancient world, ranging from Rome to China, diffused from Mesopotamia. The Mesopotamian numeration system was based on 60 as well as 10, and scholars can trace this division through many different languages. The most notable reflections of this system today are in the divisions of hours, minutes, and seconds in time calculations and in the divisions of degrees, minutes, and seconds for angle measurements. The break at 10 in Mesopotamian notation was purely additive (that is, three tens is shown by repeating the tens symbol three times), as in many simple and inconvenient numeration systems, such as the Egyptian and Greek; but the break at 60 represented true place value, making the Mesopotamians one of the four cultures that developed place value (along with the Chinese, Indian, and Maya). Some evidence suggests, however, that the Chinese and Indian place-value systems were influenced by diffusion from the Mesopotamian system, although it is also possible that the Indian system developed from diffusion from China. In any case, the Mesopotamians lacked one essential for a modern place-value system: They did not have zero. A symbol for zero was probably invented either in Indochina or India about the seventh century CE. Zero was also invented independently by the Maya, probably a hundred or so years earlier than the Indian invention, but it did not have a chance to spread around the world.

## Mesopotamia’s developing a practical method for finding square roots

Even without zero, the Mesopotamian place-value system provided many benefits, including simple algorithms for the basic arithmetic operations. Furthermore, the Mesopotamians made the logical step of extending the places to numbers smaller than one, just as we do with decimals. Sexagesimal fractions are just as convenient as decimal fractions. They contributed to Mesopotamia’s developing a practical method for finding square roots, essentially the same method once taught in elementary

and secondary schools in the United States, but now replaced with the use of electronic calculators.

## The most skilled algebraists of the ancient world

Mesopotamian mathematicians were the most skilled 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.

## Mesopotamians were the earliest people to know the Pythagorean theorem

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, sincke 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). The Mesopotamians also possessed all the theorems of plane geometry that the Greeks ascribed to Thales, including the theorem of Thales: An angle inscribed in a semicircle is a right angle. It seems unlikely, however, that these mathematicians proved their theorems from first principles, as Thales is said to have done.

## The criticism of Mesopotamian geometry

Probably the criticism of Mesopotamian geometry began because some writings seemed to use a value of three for π, a value that made its way into the Bible (indirectly –– a circular bowl is described as having a circumference three times its diameter). Later discoveries, however, have shown that at least some Mesopotamians used 3.125 for π, about as good a value as their contemporaries in Egypt had.