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, although Shamshi-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 Babylonian Period.
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 of Babylon 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 mainly Seleucid 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 the square 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 non-zero digits. These symbols and their values were combined to form a digit in a sign-value notationRoman numerals; for example, the combination 
represented the digit for 23 (see table of digits below). A space was left to indicate a place without value, similar to the modern-day zero. Babylonians later devised a sign to represent this empty place. They lacked a symbol to serve the function of radix point, so the place of the units had to be inferred from context : could have represented 23 or 23×60 or 23×60×60 or 23/60, etc. way similar to that of
Their system clearly used internal decimal to represent digits, but it was not really a mixed-radix system of bases 10 and 6, since the ten sub-base was used merely to facilitate the representation of the large set of digits needed, while the place-values in a digit string were consistently 60-based and the arithmetic needed to work with these digit strings was correspondingly sexagesimal.
The origins of Babylonian philosophy can be traced back to early Mesopotamian wisdom, which embodied certain philosophies of life, particularly ethics. These are reflected in Mesopotamian religion and in a variety of Babylonian literature in the forms of dialectic, dialogs, epic poetry, folklore, hymns, lyrics, prose, and proverbs. These different forms of literature were first classified by the Babylonians, and they had developed forms of reasoning both rationally and empirically. [3]
Esagil-kin-apli's medical Diagnostic Handbook written in the 11th century BC was based on a logical set of axioms and assumptions, including the modern view that through the examination and inspection of the symptoms of a patient, it is possible to determine the patient's disease, its aetiology and future development, and the chances of the patient's recovery.[4]
During the 8th and 7th centuries BC, Babylonian astronomers began studying philosophy dealing with the ideal nature of the early universe and began employing an internal logic within their predictive planetary systems. This was an important contribution to the philosophy of science.[5]
It is possible that Babylonian philosophy had an influence on Greek, particularly Hellenistic philosophy. The Babylonian text Dialog of Pessimism contains similarities to the agonisticsophists, the Heraclitean doctrine of contrasts, and the dialogs of Plato, as well as a precursor to the maieutic Socratic method developed by Socrates.[6] The Ionian philosopher Thales had also studied in Babylonia. thought of the
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