## The History of Mathematics## Mesopotamian Numbers and Mathematics |

## What do we know about Babylonian mathematical tables? |

Archeologists know that the Babylonians invented tables to represent various mathematical calculations. Evidence comes from two tablets found in 1854 at Senkerah on the Euphrates River (dating from 2000 B.C.E.; some researchers say 1800 to 1600 B.C.E.). One listed the squares of numbers up to 59, and the other the cubes of numbers up to 32 (for more about the second tablet, see above).

The Babylonians also used a method of division based on tables and the equation *a/b = a* × (1/b). With this equation, all that was necessary was a table of reciprocals; thus, the discovery of tables with reciprocals of numbers up to several billion.

They also constructed tables for the equation n^{3} + n^{2}, in order to solve certain cubic equations. For example, in the equation *ax ^{3} + bx^{2} = c* (note: this is in our modern algebraic notation; the Babylonians had their own symbols for such an equation), they would multiply the equation by

*a*

^{2}, then divide it by

*b*

^{3}to get (

*ax*/

*b*)

^{3}+ (

*ax/b*)

^{2}=

*ca*

^{2}/

*b*

^{3}. If

*y = ax/b,*then

*y*

^{3}+

*y*

^{2}=

*ca*

^{2}/

*b*

^{3}—and could now be solved by looking up the

*n*+

^{3}*n*table for the value of

^{2}*n*that satisfies

*n*+

^{3}*n*=

^{2}*ca*When a solution was found for

^{2}/b^{3}.*y*, then

*x*was found by

*x*=

*by/a*. And the Babylonians did all this without the knowledge of algebra or the notation we are familiar with today.