Illustrated Commentary on Nine Chapters, Vol. 1

...through movement and variation, they are unequal. Therefore, those who practice calculation must first organize the various fractions.

The method says: If a number can be halved, halve it. If it cannot be halved, set aside the denominator and the numerator. Use the lesser to subtract from the greater, and continue this mutual subtraction to find the equal number the greatest common divisor. Use this equal number to reduce the fraction. Using the equal number to reduce the fraction is the same as dividing. The reason for the mutual subtraction is that all remainders are multiples of the equal number, so one reduces by the equal number.

The calculation says: Place the denominator 18 in the lower position and the numerator 12 in the upper position. Set aside a copy. Use the upper to subtract from the lower; the lower position remains 6. Use the remaining 6 to subtract from the copied upper; the upper position also remains 6. Thus, 6 is the equal number. Use the equal number 6 to reduce 18, which gives 3. Use the equal number 6 to reduce 12, which gives 2. This results in 2/3, which matches the answer.

The calculation says: Place the denominator 91 in the lower position and the numerator 49 in the upper position. Set aside a copy. Use the upper to subtract from the lower; the lower position remains 42. Use 42 to... upper position?

Square Fields