# Sun Zi

### Quick Info

China

China

**Sun Zi**was a Chinese mathematician who wrote

*Sunzi suanjing*(Sun Zi's Mathematical Manual).

### Biography

Nothing is known about**Sun Zi**except his text

*Sunzi suanjing*(Sun Zi's Mathematical Manual). Dating this is made more difficult since it is not known how much the text was changed or added to over time. Let us first look at the various theories about the date.

In the 17th century Sun Zi was identified with Sun Wu, a famous military expert of the sixth century BC who wrote

*Sun Zi's art of war*. The Ruan Yuan in his

*Chouren zhuan*or

*Biographies of astronomers and mathematicians*(1799) certainly realised that references in certain problems in the

*Sunzi suanjing*meant that the identification with Sun Wu was incorrect. He placed Sun Zi around 250 BC but knew that there was still problems with this dating which he said would have to be studied later. Indeed such studies did take place and Dai Zhen, an 18th century scholar, historian and mathematician, stated that it was impossible for the

*Sunzi suanjing*to have been written before about 50 BC.

In more recent times Alexander Wylie, the expert on China who worked with the London Missionary Society in Shanghai for many years, states in a text published in 1897 that:-

During the third century Sun Zi, an author of considerable note, published his Sunzi suanjing.L E Dickson, the American number theorist, claimed in 1919 that the

*Sunzi suanjing*was written in the first century AD. Wang Ling [10] seems to have the most convincing argument:-

The Sunzi suanjing mentions the mein as an item of taxation, and the hu tiao system. These two were first established in 280 AD. So the book could not have been written before this date. ... A new scale between chih and tuan was established in 474 AD; the Sunzi, still using the old scale by Wu Ch'en-Shih's emendation, cannot be older than 473 AD.Of course this dating assumes that the text was written as a whole, while it seems more likely that it was compiled, like many of the texts, from older sources. In that case Wang Ling's dating will only establish when part of the text was written, some possibly being earlier, while other parts probably have been written later. This theory is supported by the fact that in a bibliography compiled in 636 it is stated that the

*Sunzi suanjing*has two chapters while today's version has three. Rather strangely it is the first of the three chapters which has a different form to the other two, so perhaps an elementary introduction was added at a later date.

What of the modern texts? Lam and Ang in [1] suggest it is a '3rd century AD treatise'; Martzloff in [2] (also [3]) gives 'fifth century very approximately'; Bag and Shen [4] say 'About 400 AD a Chinese mathematician, Sun Zi, took up the problem ...'; Berezkina [5] says it was 'composed in the third or fourth century of our era'; Lam in [6] says 'Sun Zi (somewhere between the 3rd and 5th centuries AD) ...'; while Shen [9] gives the very precise date for the

*Sunzi suanjing*of 237 AD.

Leaving the question of the date let us look briefly at the content of the treatise, before finally trying to make some guesses about Sun Zi based purely on the text. The

*Sunzi suanjing*consists, as we have already noted, of three chapters. The first chapter describes systems of measuring with considerable detail, and gives instructions on using counting rods to multiply, divide, and compute square roots. It also gives two systems for designating high powers of ten. Sun Zi explains how to do multiplication on a counting board:-

Lay down the multiplicand in the upper row and the multiplier in the lower, with the product in between. Attention must be paid to the placing of the digits.So the numbers to be multiplier are placed in the top and bottom of the three rows of the counting board and multiplications by single digits and additions take place in constructing the product in the middle row. Sun Zi then explains how to do division on a counting board:-

In division, reverse the order by placing rods in rows for quotient (upper), dividend (middle), and divisor (lower) respectively,in this case the if one were dividing 1813 by 49, then 1813 is placed in the middle row, 49 in the bottom row. Then division by single digits takes place, building up the answer in the top row, with subtraction from the middle row until it is empty.

The second and third chapters consist of problems (28 problems and 36 problems respectively) concerning fractions, areas, volumes etc. similar to, but rather easier than, the problems in the Nine Chapters on the

*Mathematical Art*. Here is a sample, rather easy, problem:-

Problem 3.34: Suppose that, after going through a town gate, you see 9 dykes, with 9 trees on each dyke, 9 branches on each tree, 9 nests on each branch, and 9 birds in each nest, where each bird has 9 fledglings and each fledgling has 9 feathers with 9 different colours in each feather. How many are there of each?Answer: 81 trees, 729 branches, 6561 nests, 59049 birds, 531441 fledglings, 4782969 feathers, 43046721 colours.

One problem, however, is of special interest, this being Problem 26 in Chapter 3:-

Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 3 are left over, and when counted in sevens, 2 are left over. How many objects are there?This, of course, is important for it is a problem which is solved using the Chinese remainder theorem. It is the earliest known occurrence of this type of problem. In fact the solution given, although in a special case, gives exactly the modern method. After solving the particular problem (the answer is 23) the

*Sunzi suanjing*gives a method for arbitrary remainders:-

Multiply the number of units left over when counting in threes by 70, add to the product of the number of units left over when counting in fives by 21, and then add the product of the number of units left over when counting in sevens by 15. If the answer is 106 or more then subtract multiples of 105.Let us mention one further major contribution by Sun Zi. Xu, in [11], gives a detailed description of the algorithm used by Sun Zi for the extraction of roots and compares it with the method described in the Nine Chapters on the

*Mathematical Art*. Xu argues convincingly that Sun Zi's procedure differs from the earlier method in significant ways and so should be recognized as an outstanding and original contribution.

Can we deduce anything of Sun Zi himself? Perhaps the most significant fact is that nothing is known. How can this tell us anything? Well first notice that since the work of modern historians has placed the

*Sunzi suanjing*much later than was thought in ancient times, we can now see that as nothing was known of Sun Zi within say 100 years of his death. For example in the

*Standard History of the Sui dynasty*the treatise is mentioned with no details of the author, and the same occurs in other similar works. Unlike many Chinese mathematicians, Sun Zi cannot have been an important government official, nor to have been from a family of high social standing. The best guess would be that he was a scholar, clearly interested in the social issues of the day, but playing a very minor role in society. One problem suggests that he may have been a Buddhist for he mentions a Buddhist sutra in Problem 4 of Chapter 3.

### References (show)

- L Y Lam and T S Ang,
*Fleeting footsteps : Tracing the conception of arithmetic and algebra in ancient China*(River Edge, NJ, 1992). - J-C Martzloff,
*A history of Chinese mathematics*(Berlin-Heidelberg, 1997). - J-C Martzloff,
*Histoire des mathématiques chinoises*(Paris, 1987). - A K Bag and K S Shen, Kuttaka and qiuyishu,
*Indian J. Hist. Sci.***19**(4) (1984), 397-405. - E I Berezkina, Le traité mathématique de Sun-zi Suan-Jing (Russian),
*Istor.-Mat. Issled.***13**(1960), 219-230. - L Y Lam, A Chinese genesis : rewriting the history of our numeral system,
*Arch. Hist. Exact Sci.***38**(2) (1988), 101-108. - L Y Lam, Correction for the article: 'A Chinese genesis : rewriting the history of our numeral system',
*Arch. Hist. Exact Sci.***39**(4) (1989), i. - W M Li, A preliminary proof of Zi Sun's theorem and the art of solving congruences by Zong Xian Huang (Chinese), in
*Di Li (ed.), Collected research papers on the history of mathematics***3**(Chinese) (Hohhot, 1992), 112-116. - K S Shen, Historical development of the Chinese remainder theorem,
*Arch. Hist. Exact Sci.***38**(4) (1988), 285-305. - L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in
*Proc. Tenth Internat. Conf. History of Science, 1962*(Paris, 1964), 489-492. - X T Xu, Sun Zi suan jing ( Master Sun's arithmetical manual) was the original source for the 'leap forward and regress method of place determination' in the extraction of roots (Chinese),
*J. East China Norm. Univ. Natur. Sci. Ed.*(1) (1987), 22-27.

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update December 2003

Last Update December 2003