Sun-tzi (? ~ ?)

Following the Han period lived the mathematician Sun-tzi, who wrote a book containing much material similar to that of the Arithmetic in Nine Sections. It is in this work that we encounter the first Chinese problem in indeterminate analysis : “There are things of an unknown number which when divided by 3 leave 2, by 5 leave 3 and by 7 leave 2. What is the smallest number?” Here we find the beginnings of the famous Chinese Remainder Theorem of elementary number theory.

Wang Fan (? ~ ?)

In the post-Han period we also find a number of mathematicians devoting attention to the computation of pi, the ratio of the circumference to the diameter of a circle. A general of the third century, named Wang Fan has been credited with the rational approximation 142/45 for pi, yielding pi  =  3.155 . A contemporary of Wang Fan,  named Liu Hui, wrote a short commentary on the Arithmetic in Nine Sections called the Sea Island Manual. In this we find some new material on mensuration, among which is the relation 3.1410 < pi < 3.1427.

Tsu Ch’ung-chih (430 AD-501AD)

About two centuries later, Tsu Ch’ung-chih and his son whose joint book is now lost, found that 3.1415926 < pi < 3.1415927 and the remarkable rational approximation 355/113 which used pi correct to six decimal places. This rational approximation was not rediscovered in Europe until 1585. The precision of pi achieved by Tsus seems not to have been surpassed until 1429, when the astronomer Jashid Al-Kashi of Samarkand found pi correct to sixteen decimal places. Western mathematicians did not surpass the Tsus approximation until around 1600s.

Ch’in Kiu-shao,   Li Yeh, Yang Hui, Chu Shi-kie

The latter part of the Sung Dynasty through the early part of the Yuan Dynasty marks the greatest period in ancient Chinese mathematics. Many important mathematicians flourished and many worthy mathematical books appeared. Among the mathematicians were Ch’in Kiu-shao, Li Yeh, Yang Hui, and the greatest of all, Ch’in Shi-kie.

Ch’in took up indeterminate equations where Sun Tzi  had left off. He was also the first Chinese to give a separate symbol, “0” for zero. He was one of the mathematicians who generalized the method of extracting square roots (as given in the Arithmetic in Nine Sections) to equations of higher degrees, leading to the numerical method of solving algebraic equations we today refer to as Horner’s method, since it was independently found by the English schoolmaster, William George Horner (1786-1837) and published by him in 1819. He was completely unaware of the fact that he had rediscovered an ancient Chinese computational scheme.

Li Yeh is of special interest in that he introduced a notation for negative numbers by placing a diagonal stroke through the right hand digit when the number is written in the Chinese scientific or rod, system.

Yang Hui, whose books are a sort of extension of the Arithmetic in Nine Sections, worked deftly with decimal fractions by essentially our present methods. Yang Hui has also given us the earliest extant presentation of the so-called Pascal arithmetic triangle, which is again in the later book written by Chu Shi-kie in 1303.

Chu Shi-kie speaks of the triangle as already ancient at his time. It would appear, then, that the binomial theorem was known in China for a long time. Chu’s books give the most accomplished presentation of Chinese arithmetic-algebraic methods that has come down to us. He employs familiar matrix methods of today, and his method of elimination and substitution has been compared to that of J.J. Sylvester (1814-1897).