Babylonian Arithmetic


For the Babylonians, addition and subtraction were very much as they are for us today except that instead of carrying 10 s the Babylonians carried 60 s.

Multiplication relied on the distributive law.

For example: 25 2 = (20 + 5) 2 = (20 2) + (5 2).

Division was accomplished by multiplying by reciprocals. For example: 47/3 = 1/3 47.

To aid such calculations, the Babylonians constructed tables of reciprocals. Generally, tables of reciprocals were only constructed for regular sexagesimal numbers. These are numbers which can be written as a product of twos, threes and fives and hence their reciprocals have a finite sexagesimal expansion. Fractions such as 1/7 (the number 7 is not regular) were either avoided or approximated.

The Babylonians also developed a method for approximating square roots. For example, to find :
      is a little more than 5.
     Now = 26 and 5 26/5 = 26
     and since 5 is a little less than , 26/5 is a little greater than .
     So to approximate the Babylonians took the average approx 1/2(5 + 26/5) = 51/10.
In a similar but iterative method, they obtained approx 1/2(3/2 + 4/3) = 17/12.

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