Chapter 15 Section 6 Surplus and Deficit Technique

The problem of surplus and deficiency constitutes the seventh chapter of "Nine Chapters".Its typical question is: Today there is a total purchase, people pay for a, surplus b, people pay for a, shortage of b, how much is the number of people and the price? The first method in "Nine Chapters" is: "Set the output rate, the surplus and the deficiency are at the bottom. Make the dimension multiplied by the output rate, and take it as a reality, and combine the surplus and insufficiency as the law, and the reality is one like the law."This is to find the actual ab+ab, the method b+a, assuming that the number of people to be sought is u, and the price is v, then

v/u=(ab+ab)/(b+b) (1) It is the number of no profit and no loss that each person should pay.Regarding the problem of co-purchasing goods, "Set the rate of output, reduce the excess with less, and use the law to make the surplus. The actual price is the price, and the law is the number of people", that is u=(b+b)/( | aa | ) v=(ab+ab)/( | aa | ) Liu Hui proved its correctness with the principle of homogeneity.This is to first share the profit and deficiency as bb, and the output rate must be equal to the surplus and deficiency, so that it becomes ab, ab, and the problem becomes that if a is produced for b times, the total profit is bb, and a is produced for b times, and the total deficiency is bb.Therefore, if ab+ab is issued b+b times in total, there will be no profit or loss, and each output will be (ab+ab)/(b+b).And b+b is the difference between all people, it is accumulated from the difference | aa | of one person, so (b+b)/( | aa | ) is the number of people, which also proves the second kind of "Nine Chapters" correctness of the method.The second method gives the formula u=(b+b)/( | a+a | ), v=ua-b=ua-b. "Nine Chapters" also gives the problems of two surpluses, two deficiencies, surplus and insufficiency

For any arithmetic problem, assuming an answer and substituting it into the original problem for checking, one of the three situations of surplus, deficiency, and adequacy will inevitably occur. If it is assumed twice, it becomes a surplus-deficiency problem. Formula (1) is for these problems Proposed.Take the question of "oil self and lacquer" as an example.It is known that paint 3 can be changed with oil 4, and oil 4 can be adjusted with paint 5.Existing lacquer 3 buckets, want to take out a part to change oil, make the oil that changes just can reconcile remaining lacquer.Ask about the paint used for oil change, the oil obtained, and the paint to be blended?The solution is: if the paint used for oil change is 9 liters, then 12 liters of oil can be exchanged, 15 liters of adjustable paint, 30-(9+15)=6, less than 6 liters; 12 liters, then get 16 liters of oil in exchange, 20 liters of adjustable paint, (12+20)-30=2, there is a surplus of 2 liters.Substituting (1) formula, the paint that is used for oil change is (12 * 6+9 * 2)/(2+6)=111/4 (liter), changes 15 liters of oils, and the lacquer of blending is 183/4 liters.

In the early days of the development of Chinese mathematics, this method of two assumptions was often used to solve complex problems by turning them into surplus-deficit problems.This method can give exact answers to linear problems, but only approximate solutions to nonlinear problems, which the author of "Nine Chapters" did not realize.For example: There is a wall with a thickness of 5 feet. Two mice cross each other. On the first day, they both wear 1 foot. From the next day, the number of rats doubles and the number of mice decreases by half. When will the two mice meet again? The solution of "Nine Chapters" is: if the order is 2 days, it is less than 5 inches, if the order is 3 days, there is a surplus of 3 feet 71/2 inches, and it is substituted into (1) to get 2 (2/17) days.But this question is nonlinear, and the exact solution should be lg(2+√6)/lg2.However, even in advanced mathematics, it is an effective method to solve complex problems with the technique of surplus and deficiency, such as the method of borrowing or the method of chords to find the root of f(x)=0, the principle of which is the technique of surplus and deficiency .

After being introduced to Arabia and the West, the technique of surplus and deficiency has long been their main method for solving mathematical problems.The Arabs call it the Khitan algorithm, also known as the double method.