Chapter 7 of the Nine Chapters is devoted to the use of a technique called ying bu tsu shu (literally “the rule of too much and not enough”), often translated as “the method of excess and deficit”. The technique amounts to a way of dealing with linear relationships without the use of algebra. Chinese mathematicians were ingenious in applying it to solve many different types of problems, even nonlinear problems solved by linear approximation.
The thinking behind ying bu tsu shu was influenced by observations based on work with fractions. In problems such as those involving the addition or subtraction of fractions, the denominators have to be reconciled first; for example:
\[{\frac{8}{3}}+{\frac{7}{4}}={\frac{4\cdot 8}{4\cdot 3}}+{\frac{3\cdot 7}{3\cdot 4}}={\frac{4\cdot 8+3\cdot 7}{4\cdot 3}}={\frac{53}{12}}\]
The Chinese used the term tong (“uniformization”) for the process of creating a common denominator, such as \(4\cdot 3.\) They used the term qi (“homogenization”) for the cross-multiplication that is needed to compare, and in this case add, the numerators: \(4\cdot 8+3\cdot 7.\)
To see how the Chinese adapted homogenization to solve for unknown quantities in linear problems, consider the first example from Chapter 7:
Now an item is purchased jointly; everyone contributes 8 [coins], the excess is 3; everyone contributes 7, the deficit is 4. Tell: The number of people, the item price, what is each? (Shen et al., p. 358)
In ancient times, when algebra as we know it did not exist, this joint purchase problem was not at all easy to solve. Here is how the Chinese viewed its solution, based on Liu Hui’s written commentary from 263 AD. We are given:
8 coins per person \(\leftrightarrow\) 1 item and 3 more coins
7 coins per person and 4 more coins \(\leftrightarrow\) 1 item.
By quadrupling or tripling, respectively, we get
\(4\cdot 8\) coins per person \(\leftrightarrow\) 4 items and \(4\cdot 3\) more coins
\(3\cdot 7\) coins per person and \(3\cdot 4\) more coins \(\leftrightarrow\) 3 items.
Adding the results,
\(4\cdot 8\) + \(3\cdot 7\) coins per person \(\leftrightarrow\) 4 + 3 items
Taking the ratio,
\(\displaystyle{\frac{4\cdot 8+3\cdot 7}{4 + 3}}\) coins per person \(\leftrightarrow\) 1 item
\(\displaystyle\frac{53}{7}\) coins per person \(\leftrightarrow\) 1 item.
So, to afford the purchase, each person must contribute \(\displaystyle\frac{53}{7}\) the value of one coin. Note the cross-multiplication used in the calculation. This was a pattern that the Chinese committed to memory for use as a shortcut, much as in the problem \(\displaystyle{\frac{8}{3}}+{\frac{7}{4}}\).
Once the true cost per person is known, the solution is easily finished. Comparing each person’s true cost of \(\displaystyle\frac{53}{7}\) coins to the \(8=\displaystyle\frac{56}{7}\) coins per person that led to an excess of 3 coins, we see that those 3 coins resulted from an overpayment of \(\displaystyle\frac{3}{7}\) coin per person. Thus, there must be 7 buyers involved. Finally, 7 buyers each contributing \(\displaystyle\frac{53}{7}\) coins implies a total price of 53 coins for the item.
The British scholar Joseph Needham (1900-1995), one of the leading historians of Chinese mathematics and science, pointed out that the rationale of balancing excess and deficit found in the Nine Chapters seems to reflect one of the key doctrines of Confucius (c. 551–479 BCE), whereby yin and yang must be balanced to achieve harmony (Needham, p. 119). In medieval times, the Arabs developed, and introduced to Europe, a technique called double false position that is somewhat similar to the Chinese method of excess and deficit. However, double false position was based on Greek theories of ratio and proportion, rather than on Confucian doctrines of homogenization and balance.
Here is the next problem from Chapter 7. Can you solve it?
Problem 6. Now chickens are purchased jointly; everyone contributes 9, the excess is 11; everyone contributes 6, the deficit is 16. Tell: The number of people, the chicken price, what is each? (Shen et al., p. 358)
Download solutions to Problems here: Students Explore the Nine Chapters from China.