They invented and developed arithmetic by using several different number systems including a mixed radix system with an alternating base 10 and base 6. This sexagesimal system became the standard number system in Sumer and Babylonia. They may have invented military formations and introduced the basic divisions between infantry, cavalry, and archers. They developed the first known codified legal and administrative systems, complete with courts, jails, and government records. The first true city-states arose in Sumer, roughly contemporaneously with similar entities in what are now Syria and Lebanon. Several centuries after the invention of cuneiform, the use of writing expanded beyond debt/payment certificates and inventory lists to be applied for the first time, about 2600 BC, to messages and mail delivery, history, legend, mathematics, astronomical records, and other pursuits. Conjointly with the spread of writing, the first formal schools were established, usually under the auspices of a city-state's primary temple.

Sumerians used number systems which included what alternating bases?