The Babylonian Tables

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Who were the Babylonians?

The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers. They developed an abstract form of writing based on cuneiform (wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn.

Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation.

Babylonians and 60

The Babylonians had an advanced number system, in some ways more advanced than our present system. It was a positional system with base 60 rather than the base 10 of our present system. Now 10 has only two proper divisors, 2 and 5. However 60 has 10 proper divisors so many more numbers have a finite form.

The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write  5 hours, 25 minutes, 30 seconds is just to write the base 60 fraction, 5 25/60 30/3600 or as a base 10 fraction 5 4/10 2/100 5/1000 which we write as 5.425 in decimal notation.

One major disadvantage of the Babylonian system however was their lack of a zero. This meant that numbers did not have a unique representation but required the context to make clear whether 1 meant 1, 61, 3601, etc.

Multiplication

The Babylonians used the formula ab = ((a + b)²  - a² - b² )/2 to make multiplication easier. Even better is the formula ab = (a + b)² /4 - (a - b)² /4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of two numbers that were looked up in the table.

Division

Division is a harder process. The Babylonians did not have an algorithm for long division. Instead the based their method on the fact that a.b = a.(1/b) so what was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion.

Now the table had gaps in it since 1/7, 1/11, 1/13, etc. do not have terminating base 60 fractions. This did not mean that the Babylonians could not compute 1/13, say. They would write 1/13 = 7/91 = 7.(1/91) =(approx.) 7.(1/90) and these values were given in the tables.

Triangles

One of the Babylonian tablets (Plimpton 322) which is dated from between 1900 and 1600 BC contains answers to a problem containing Pythagorean triples, i.e. numbers a, b, c with a²  + b²  = c² . It is said to be the oldest number theory document in existence. You can see a picture of this tablet on the right.

A translation of another Babylonian tablet which is preserved in the British museum goes as follows

4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.