Numbers, Mesopotamian

Abstract

One must distinguish the abstract concept of number from the use of a symbol representing the counted item. In Mesopotamia, the use of counting symbols, in the form of stone tokens, is attested in the seventh millennium BCE, well before the use of writing. By the time writing appears in URUK at the end of the fourth millennium, approximately a dozen different numerical systems were in use. Each system was associated with a certain type of object or product. In other words, the number signs give information both on a number of units and on the type of object considered (Nissen, Damerow, and Englund 1993).The third millennium witnesses improvements in differentiating numerical signs, from metrological signs, for example, the development during the UR III DYNASTY of a system of weight-measures where the numbers -taken from the system used to count discrete objects -are accompanied by the name of the corresponding weight unit. There is also an attempt to reduce the number of systems.At the end of the third millennium or the beginning of the second, the concept of abstract number appears in the form of a sexagesimal floating-place-value notation. However, at this time, this system is found exclusively in school mathematical texts, but this same system would also be used a thousand years later in the astronomical texts. The administrative texts continued to use the traditional systems related to the counted item, written in sign value notation -that is, in which each sign is given a value, for example to double the value, the sign is written twice, etc.In the Babylonian sexagesimal place-value notation, the numbers one to fifty-nine are written decimally using two digits: one ( ) and ten ( ); each step to the left of a double digit increases its value by a power of sixty. For example, the number 5563 in the Babylonian place value notation (1 Â 60 2 þ 32 Â 60 1 þ 43 Â 60 0 ) is written:. The same number in the traditional notation for discrete objects is written: 3600 þ (3 Â 600) þ (2 Â 60) þ (4 Â 10) þ (3 Â 1):where represents 3600; represents 600; represents 10; represents 1 (or 60).The Babylonians of the second millennium BCE did not have a number zero or a sign denoting a final zero. The consequence is that the magnitude of the number is not fixed in the place value notation, this is why this system is called "floating." However, the use the Babylonian scribes made of this system shows that they had no need for a zero (Proust 2007;2008). This system is used, for example, to learn multiplication and division and to perform abstract calculations in order to solve a problem often given in the traditional metrological system. Hence, the magnitude was already known, and the sexagesimal placevalue notation is used as a calculation tool.