**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 17^{th} 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 18^{th} 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:-

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

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

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 theThe Sunzi suanjing mentions the mein as an item of taxation, and the hu tiao system. These two were first established in280AD. So the book could not have been written before this date. ... A new scale between chih and tuan was established in474AD; the Sunzi, still using the old scale by Wu Ch'en-Shih's emendation, cannot be older than473AD.

*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 '3^{rd} 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 3^{rd} and 5^{th} 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:-

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:-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.

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.In division, reverse the order by placing rods in rows for quotient(upper), dividend(middle), and divisor(lower)respectively,

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:-

Answer: 81 trees, 729 branches, 6561 nests, 59049 birds, 531441 fledglings, 4782969 feathers, 43046721 colours.Problem3.34: Suppose that, after going through a town gate, you see9dykes, with9trees on each dyke,9branches on each tree,9nests on each branch, and9birds in each nest, where each bird has9fledglings and each fledgling has9feathers with9different colours in each feather. How many are there of each?

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

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) theSuppose we have an unknown number of objects. When counted in threes,2are left over, when counted in fives,3are left over, and when counted in sevens,2are left over. How many objects are there?

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

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.Multiply the number of units left over when counting in threes by70, add to the product of the number of units left over when counting in fives by21, and then add the product of the number of units left over when counting in sevens by15. If the answer is106or more then subtract multiples of105.

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.

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