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Babylonia - Base 60

Babylonian Counting and Mathematics


Table of Squares

Table of Squares

NS Gill

The Babylonians used a sexagesimal (base 60) system that was so functional that today, 4000 years later, we still use it every day -- whenever we tell time or refer to degrees of a circle.

One of the main differences between our system and theirs is the number of factors; 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 for base sixty and 1, 2, 5, and 10 for base ten. This impresses me because I remember the difficulty I had learning multiplication tables. The only rows I learned effortlessly were the factors and the diagonal (the squares). A system with so many more factors should be correspondingly easy to learn.

On the other hand, I could fault the Babylonians for having developed those odious tables in the first place, but then they didn't develop times tables. Instead, they multiplied using a formula that depended on knowing only the squares. With only their table of squares (albeit going up to a monstrous 59 squared), they could compute the product of two integers, a and b, using a formula similar to:

ab = [(a + b)2 - (a - b)2]/4

The main fault of the Babylonian system was the absence of a zero. But the Ancient Maya's vigesimal (base 20) system had one, drawn as a shell. Other numerals were lines and dots, similar to what we use today when we tally.

Because of their mathematics, the Babylonians and Maya had elaborate and fairly accurate measurements of time and the calendar. Today, with the most advanced technology ever, we still have to make temporal adjustments -- almost 25 times a century to the calendar and a few seconds every few years to the atomic clock.

There's nothing inferior about modern math, but the next time my friend rants about her son's refusal to learn the times table, I will suggest she present the boy a Babylonian alternative.

Online Resources

Links to pages on the internet that explain the history of and the variety in calendars.

Babylonian and Egyptian Mathematics
In addition to a more complete explanation of Babylonian tables, this site shows the Babylonians knew what came to called the Pythagorean theorem. Egyptian and Greek math may have been similar but one was practical and the other theoretical.

Written by N.S. Gill 07/01/97

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