By 3000 B.C., these marks had evolved into complex systems of numbers, capable of recording very large quantities of goods. The number systems were still metrological in nature. That is, a scribe used different collections of signs for recording quantities of sheep or grain. More than a dozen different such systems are known and in each case, the ratio between the quantity denoted by one sign and the quantity denoted by the 'next biggest' sign was different. There was no standardized base, just as nowadays we still have no standardized base for all our different units of weights and measures outside of the metric system.
Over the next 500 years, writing gradually developed into the cuneiform script. Cuneiform means "wedge-shaped" and refers to the way scribes wrote marks on wet clay, using the cut tip of a stylus. You can find out a great deal more about cuneiform and the Akkadian language on John Heise's Akkadian pages. Along with the emergence of cuneiform came the emergence of arithmetic. Clay tablets were no longer used merely as places to record numbers of goods, but we find computations of totals and calculations of areas.
By the Old Babylonian period of around 2000 B.C. there was a fully-developed mathematics. Thousands of mathematical and economic tablets have been recovered. They detail an impressive knowledge of arithmetic, a great facility with what we would consider as linear and quadratic equations, numerous geometrical constructions and computations. There are multiplication tables, tables of squares, square roots, reciprocals, common constants. There are lists of problems for teachers to set, and solutions given by students. At this time, the number system for computation had settled down into a sexagesimal, or base sixty, place-value system, much like our own, except that it was easier for them to compute fractions. However, since most mathematics was applied, problems tended to be stated, and answers had to be given in particular units of weights and measures. Students had to learn how to convert from these metrological systems into sexagesimal and then back again at the end. There are many tablets giving conversion constants.
Babylonian mathematics does not seem to have changed much in the next
1500 years. Partly, this seeming may be due to the fact that we
have very few mathematical tablets from this period. The next great flowering
we know of was the development of mathematical astronomy in the late Babylonian
period of the last few centuries B.C.