Zhang Qiujian

Born: about 430 in China
Died: about 490 in China

Zhang Qiujian's name is sometimes written as Chang Ch'iu-Chin or Chang Ch'iu-chien. Zhang Qiujian wrote his mathematical text Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) some time between 468 and 486. No other work by Zhang Qiujian is known, and nothing about its author is known. In fact. although the dats we give are based on the best available evidence, some authors place Zhang Qiujian 100 years earlier than we have done. At least the reader should be aware of this lack of knowledge. This article therefore can amount to nothing more than a look at the text and see if any deductions about the author can be made from looking at his treatise.

The book comprises of three chapters, with 32 problems in the first chapter, 22 problems in the second, and 38 problems in the third chapter. Each problem is followed by an answer and many are followed by a description of the method of determining the answer. However, no reasons are given for the method of solution. There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving systems of linear equations. There are problems concerned with proportions, compound proportions, and proportional parts. There are also geometric problems, which depend on knowing the relevant formulae for areas and volumes.

The treatise is written in a similar style to the Nine Chapters on the Mathematical Art but the problems which it considers are usually more advanced. The book certainly represents progress in Chinese mathematics beyond the Nine Chapters on the Mathematical Art. Much of the book is designed to give the reader practice at manipulating fractions. Zhang states in his preface:-

Anyone who studies mathematics should not be afraid of the difficulty of multiplication and division, but should be afraid of the mysteries of manipulating fractions.
If manipulating fractions is hard then the student will need practice. Zhang asks the reader to divide 12380/7 by 138/5 in Problem 5 of Chapter 1 and to calculate (6587 + 2/3+ 3/4) divided by 58 1/2(= 117/2) in the next problem. He gives the reader the same method of dividing fractions as taught in schools today, namely invert the divisor and multiply.

In particular Zhang sees the reduction of fractions to a common denominator as hard. He therefore gives problems which will help the reader. For example Problem 10 in Chapter 1 reads:-

A circular road around a hill is 325 li long. Three persons A, B, and C run along the road. A runs 150 li per day, B runs 120 li per day, and C runs 90 li per day. If they start at the same time from the same place, after how many days will they meet again.
The method Zhang gives is to find the greatest common divisor of 150, 120 and 90. This is 30. Then divide the length of the road by the greatest common divisor to get 325/30= 10 5/6 days.

In Problem 22 of Chapter 2 segments of a circle are considered. The chord of the segment is given, as is its area, and the student is asked to compute its height (the length of the perpendicular bisector of the chord to the circle). Zhang gives the solution by solving a quadratic equation, but his formulae are not particularly accurate. Martzloff [2] points out that the length as calculated by Zhang in this problem is in error by about 14%.

In Chapter 3 problems which involve solving systems of equations occur. For example Problem 4 of Chapter 3:-

There are three persons, A, B, and C each with a number of coins. A says "If I take 2/3 of B's coins and 1/3 of C's coins then I hold 100". B says If I take 2/3 of A's coins and 1/2 of C's coins then I hold 100 coins". C says "If I take 2/3 of A's coins and 2/3 of B's coins, then I hold 100 coins". Tell me how many coins do A, B, and C hold?
A modern solution sets A to have x coins, B to have y and C to have z. Then
x + 2y/3 + z/3 = 100
2x/3 + y + z/2 = 100
2x/3 + 2y/3 + z = 100
giving x = 60, y = 45, z = 30.

The method Zhang uses is essentially Gaussian elimination on the matrix of coefficients.

Problem 38 of Chapter 3 is perhaps the most famous in the whole of Zhang's treatise:-

Hundred fowls problem: Cockerels costs 5 qian each, hens 3 qian each and three chickens cost 1 qian. If 100 fowls are bought for 100 qian, how many cockerels, hens and chickens are there?
Zhang gives three possible answers.
  1. 4 cockerels costing a total of 20 qian, 18 hens costing a total of 54 qian and 78 chickens costing a total of 26 qian.

  2. 8 cockerels costing a total of 40 qian, 11 hens costing a total of 33 qian and 81 chickens costing a total of 27 qian.

  3. 12 cockerels costing a total of 60 qian, 4 hens costing a total of 12 qian and 84 chickens costing a total of 28 qian.
These are all the possible answers (one would not expect Zhang to come up with 0 cockerels, 25 hens and 75 chickens).

We do not know if Zhang had a systematic method to solve problems of this type or whether he solved them by trial and error.

In 656, after editing by Li Chunfeng, the treatise Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084.

Can we say anything about Zhang himself? Well, the text is very clearly written with good explanations. Examples are well chosen to illustrate what Zhang is trying to teach rather than for particularly practical purposes. We can certainly deduce that Zhang was a fine teacher and may well have had experience in teaching, to be able to write with so much knowledge of how students learn.

Article by: J J O'Connor and E F Robertson

December 2003
MacTutor History of Mathematics