Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), withBabylon as its capital. Babylonia emerged when Hammurabi (fl. ca. 1696 – 1654 BC, short chronology) created an empire out of the territories of the former Akkadian Empire. Babylonia adopted the written Semitic Akkadian language for official use, and retained the Sumerian languagefor religious use, which by that time was no longer a spoken language. The Akkadian and Sumerian traditions played a major role in later Babylonian culture, and the region would remain an important cultural center, even under outside rule, throughout the Bronze Age and the Early Iron Age. The earliest mention of the city of Babylon can be found in a tablet from the reign of Sargon of Akkad, dating back to the 23rd century BCE. Following the collapse of the last Sumerian "Ur-III" dynasty at the hands of the Elamites (2002 BCE traditional, 1940 BCE short), the Amorites gained control over most of Mesopotamia, where they formed a series of small kingdoms. During the first centuries of what is called the "Amorite period", the most powerful city states were Isin and Larsa, althoughShamshi-Adad I came close to uniting the more northern regions around Assur and Mari. One of these Amorite dynasties was established in the city-state of Babylon, which would ultimately take over the others and form the first Babylonian empire, during what is also called the Old BabylonianPeriod. Babylonian mathematics (also known as Assyro-Babylonian mathematics) refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall ofBabylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainlySeleucid from the last three or four centuries B.C. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to accurate to five decimal places.

Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of thesquare root of 2 in four sexagesimal figures, which is about six decimal figures. Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astronomical observations and calculations (aided by their invention of the abacus), used a sexagesimal (base-60) positional numeral systemSumerian and also Akkadian civilizations. Neither of the predecessors was a positional system (having a convention for which ‘e inherited from the nd’ of the numeral represented the units).

Babylonian numerals

This system first appeared around 3100 B.C. It is also credited as being the first known positional numeral system, in which the value of a particular digit depends both on the digit itself and its position within the number. This was an extremely important development, because non-place-value systems require unique symbols to represent each power of a base (ten, one hundred, one thousand, and so forth), making calculations difficult. Only two symbols ( to count units and to count tens) were used to notate the 59...