Near the Beginning of the First Century AD, About 2000 Years Ago, the Chinese Mathematical

CHINESE

FRACTION

REDUCING

Near the beginning of the first century AD, about 2000 years ago, the Chinese mathematical text called the Chiu Chang was written. No one knows for sure who wrote the text, which contains nine chapters of mathematical topics important to Chinese society at the time. Problems and solutions are presented in the text, and since the answers can be difficult to understand, different Chinese mathematicians over the years have supplied commentary and helped to make the problems and solutions clearer.

The first chapter, called Fang thien (Land Surveying), is mostly concerned with calculating the areas of fields (thien) using the basic unit of measurement, the fang (square unit). This chapter also discusses methods for working with fractions, including a way for simplifying (reducing) them. If you have a reducible fraction called m/n, the rule from the Chinese text for reducing m/n is this: If both numbers can be halved, then halve them. Otherwise set down the denominator below the numerator, and subtract the smaller number from the greater number. Continue this process until the common divisor, teng, is obtained. Simplify the original fraction by dividing both numbers by teng.

Here is an example from the Chiu Chang to illustrate the Chinese fraction reducing method:

Simplify the fraction 49/91.

Solution: The numbers cannot be halved, so we continue with the procedure. Set down the denominator below the numerator, then subtract the smaller number from the greater number:

49 91 subtract and get 42.

Now follow the process, subtracting the smaller number from the greater number until you reach a common divisor.

49 49 7 7 7 7 7 7 91 42 42 35 28 21 14 7 teng is 7.

The common divisor, or teng, is 7, so divide the numerator and denominator of 49/91 by 7 to get the simplified fraction 7/13.

YOUR PROJECT:

Simplify the following fractions using the Chinese fraction reducing method for finding teng. Show clearly the steps of your procedure.

1. 51/85

2. 36/144

3. 77/660

Reference: Chinese Fraction Reducing

Joseph, George Gheverghese. (1991). The Crest of the Peacock: Non- European Roots of Mathematics. London: Penguin Books.