Around 2000 B.C., a people called the Amorites invaded Sumer and captured its cities. These people became known as the Babylonians, whose civilisation lasted for a millennium and a half, until the capture of Babylon by the Persians in 538 B.C.
The Babylonians made significant advances in mathematics over previous civilisations. While retaining much of Sumerian mathematics, as well as most of the Sumerian number system, they then did something unique in the ancient world: They invented a positional number system. The Babylonians dropped most of the Sumerian symbols that were used to write numbers, and kept only two: The "wedge", which represented 1, and the "hook", which represented ten.
The Hindu-Arabic number system that we use today is also a positional system. In a positional (or place value) number system, the position of the number indicates the value attached to it. For example, the value of the "4" in 43 is 40 because it appears in the tens place. On the other hand, the value of "4" in 34 is 4, because it is in the ones place.
Our number system is a base ten system. The Babylonians used a base 60 system. Here is a brief overview of how they formed numbers: 1 was represented as a wedge, 2 as two wedges, and so on up to 9 as nine wedges, 10 as a hook, 11 as a hook and a wedge, and so on up to 59 which was represented as five hooks and nine wedges. To represent 60, a wedge was placed in the sixties place.
This system wasn't perfect. For example, there was no zero to use as a placeholder. Therefore,a number like 61 (1 × 60 + 1 × 1) would look very similar to 3601 (1 × 3600 + 1 × 1) because in the latter number, the 60's place was left blank. Around the time of Alexander the Great (more than 200 years after Babylon was captured by the Persians) the Babylonians fixed this problem by using two oblique wedges as a placeholder. Another problem was that there was no decimal point. This would make a number such as 1/2 (30 × 1/60) look the same as 30 (30 × 1). The representation of many numbers was often ambiguous, so scribes had to use the numbers' context to determine their value. Still, the invention of a positional system was a great achievement in mathematics, considering that millennia later the Greeks still used cumbersome number systems like the Attic and Ionic numerals. In Europe Hindu-Arabic numerals did not catch on until about 1500 A.D., more than three millennia after the Babylonians adopted their place value system.
The Babylonians made other significant advances in other areas of mathematics, such as fractions, algebra, and geometry. There are tiny bits about them in my pages on the Pythagorean theorem and 2.
See also: Mathematics history