The Babylonian Number System

The Babylonians used a base 60, or sexagecimal, number system which is preserved in tablets made by pressing into the soft clay styli with carved cuneiform script. Our habit of giving 360o to a circle, 60 seconds in a minute and 60 minutes in an hour are cultural artifacts passed down to us from the Babylonians. Cuneiform script is shown at the right. The symbols for 1-9 and 10 - 50 are given in the first two lines followed by examples of the decimal (subscript D) numbers 12, 37, 97, 65 and 12709 converted to their sexagecimal (subscript S) equivalents. Zero was indicated simply by a space. Information on this number system comes to us from some four hundred clay tablets containing lists of mathematical problems and mathematical tables of more than half a million engraved in their written cuneiform script which have been excavated since the first half of the nineteenth century. A particularly well studied tablet is shown below. It is believed to have been found at Senkereh and was purchased by George Arthur Plimpton in 1923 and brought to the U.S. but was not deciphered until 1945 by Neugebauer and Sachs. Designated PLIMPTON 322 in the Plimpton Collection at Columbia University, it is the oldest surviving document on number theory. It was made in Babylon between 1900 and 1600 B.C. Traces of modern glue can be seen on the left edge but the left fragment is now lost. Exhibits of these tablets can be found in the great museums of Paris, Berlin and London as well as the archeological exhibits at Yale, Columbia and the University of Pennsylvania.

Cuneiform writing was difficult to learn. The teacher of cuneiform prepared an example on the left half of a soft clay tablet and the student was instructed to copy that example on the right half.