Illustrated Commentary on Nine Chapters, Vol. 1

subtract from the copied upper 49; the upper position remains 7. Use 7 to subtract from the lower 42; the lower position also remains 7. Thus, 7 is the equal number. Use the equal number 7 to reduce 91, which gives 13. Use the equal number 7 to reduce 49, which gives 7. This results in 7/13, which matches the answer.

The commentary says: The note states, "Regarding the reduction of fractions, that the quantity of an object cannot always be expressed as a whole number" means that "whole" refers to the denominator multiplied by the whole; it means dividing by the divisor to get a whole number. "Cannot be expressed as a whole number" means the dividend does not reach the divisor; by naming it with the divisor, one has a fraction. The statement "first organize the various fractions" refers to the comprehensive categories of reduction, addition, subtraction, distribution, equalization, division, and multiplication of fractions. Zhang Qiujian a 5th-century mathematician also said: "Those who study calculation do not fear the difficulty of multiplication and division; they fear the difficulty of managing fractions."