By I. Todhunter

This quantity is made from electronic pictures from the Cornell collage Library old arithmetic Monographs assortment.

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083333. . The 3s carry on forever. But because 1/12 can also be written as 5/60, in base 60 the number 2 1/12 is written as 2;05. The sexagesimal expression is much simpler than the decimal expression of the same number and there is no round-off error. —than they are in our own decimal system. This may explain why the Mesopotamians chose the sexagesimal system. It made their computations easier to perform accurately. We should not 20 NUMBERS forget, however, that the lack of a 0 meant that the system that they employed for writing numbers with fractional parts had the same ambiguities that are to be found in their system for writing counting numbers.

Since the Mesopotamians used sexagesimal notation instead of decimal, we do not use a point. Instead we use a semicolon (;) to separate the 1s column from the 1/60 column. ) Furthermore we use a space to separate every column to the right of the semicolon. For example, the number that we might write as 1 2/60 is written as the sexagesimal number 1;02. ) To see the advantage of computing with sexagesimal notation we need only consider a common fraction such as 1 2/3. 66666. . The decimal expression does not terminate; the 6s go on forever.

The Mayans maintained two systems of numeration. One had religious significance and used a combination of multiples of 20 and multiples of 360. Although this system was important to the Mayans, Spanish accounts make clear that there was a second system of numeration in common usage. This “common system” was a base 20 system. This is the system on which we concentrate. Referring, again, to the story in the first chapter about the Malagasy system of counting, the Mayans would have placed up to 19 pebbles in the 1s pile.