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Home , Chinese mathematics, Zhoubi Suanjing

Lecture 15. of Medieval China

For a long period, the geographical nature of the country such as mountains, desserts and seas formed natural boundary which isolated China. As a result, independent of other civilizations, there was a continuous cultural development in China from ancient time and it is fascinating to trace mathematical development within that culture.

As we analysed in Lecture 2 and Lecture 14, unlike Greek mathematics there is no ax- iomatic development of mathematics in China. was very much prob- lem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes.

There are several items of important mathematical literature in the Chinese , including:

Figure 15.1 Bamboo strips of the Suan Shu Shu (left) In Olympic Game Opening Ceremony 2008 in China, ancient scholars hold bamboo strips. (right)

92 The book: Suan Shu Shu In 1984 in Province, a book Suan Shu Shu (Book of ) was discovered. It is a book written on bamboo strips and is dated around 180 B.C. It contained 69 sections of mathematical problems and solutions. For example, Section 4 is about , and Sections 61-68 are about . Some problems used a division of two fractions. For example: 2 3 2 7 14 ÷ = × = . 4 7 4 3 12

The book: An text, the Zhoubi Suanjing (Zhou Shadow Gauge Manual), was compiled between 100 B. C. and 100 A. D. It shows how to measure the positions of the heavenly bodies using shadow gauges. It also contains important sections on mathematics. The Zhoubi Suanjing contains a statement of the Gougu rule (the Chinese version of Pythagoras’s theorem): a2 + b2 = c2, and applies it to surveying, astronomy, and other topics. But the proof of the Gougu rule was not in this book. The first proof of Gougu rule was made in about 300 by a Chinese mathematician Zhao Shuang.

Figure 15.2 The Jiuzhang Suanshu

The book: Jiuzhang Suanshu The most famous Chinese mathematics book of all time is the Jiuzhang Suanshu (the Nine Chapters on the Mathematical Art). The book collected lots of mathematical results over quite a long period. There were more than one author, some of which are not known. (about 220 - about 280) who wrote his commentary on the Jiuzhang Suanshu in about 263. From his commentary, one of revisers for this book was Zhang, Chang (250-152 B.C.).

93 The Jiuzhang Suanshu collected 246 mathematical problems. The nine chapters of this book are as follows.

1. Square fields measurement (38 problems, about calculate areas).

2. Cereals and rice (46 problems, about ratio problems and trading problems).

3. Proportional distribution (20 problems, about distribution, tax, salary, sales prob- lems).

4. Given area or volume find sides (24 problems, given area of a domain, to find length of the boundary).

5. Volumes (28 problems, find volumes for many problems in architecture, and civil en- gineering).

Figure 15.3 Liu Hui and his exhaustion method in which he inscribed 3072 sides of a polygon in a circle to get π ≈ 3.14159.

6. Transportation (28 problems, about population, distance and transportation prob- lems).

7. Excess and deficient (20 problems, about profit or loss in business).

8. (18 problems, one with several unknowns).

9. Gouge Rule (24 problems, a2 + b2 = c2, and x2 + px = q).

94 Liu Hui and his approximate π by approximating polygons The earliest written approximation of π are as follows. Around 1900-1600 B.C., in Babylon, π ≈ 3.125; around 1650 B.C., in Egypt, π ≈ 3.1605; around 1200 B.C., in China, π ≈ 3; around 600 B.C., in India, π ≈ 3.088. Archimedes gave the first algorithm for calculating π by using polygons inside and outside a circle and by computing the perimeters of the polygons.

Around 263, instead of computing the segment length, Liu Hui (about 220 - about 280), who wrote his commentary on the Jiuzhang Suanshu, computed the area of the regular 192- polygon inside the unit circle to obtain 3.141024 < π < 3.142704, and the area of the regular 3072-polygons to get π ≈ 3.14159.

Liu also wrote Haidao Suanjing (Sea Island Mathematical Manual) which was originally an appendix to his commentary on the Jiuzhang Suanshu. In the book Liu uses Pythagoras’s theorem to calculate heights of objects and distances to objects which cannot be measured directly. This was to become one of the themes of Chinese mathematics.

Figure 15.4 The .

The book: Mathematical Manual of Sun Zi The book the Mathematical Manual of Sun Zi was written by Sun Zi in the late third century. In the book, it has the following problem: It is required to find an integer that leaves remainder 2 on division by 3, remainder 3 on division by 5, and remainder 2 on division by 7. Although the answer can be easily found by experiment, Sun Zi offers the following explanation which indicates a general method.

95 If we count by 3 and there is a remainder 2, put down 140. If we count by 5 and there is a remainder 3, put down 63. If we count by 7 and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer.

A method for solving Sun Zi’s problem in full generality was first given in the Mathe- matical Treatise in Nine Sections by in 1247. He solved the crucial problem of finding inverses by the Euclidean algorithm.

Chinese remainder theorem If p1, ..., pk are relatively prime integers and r1 ≤ p1, ..., rk ≤ pk are any non negative integers, then there is an integer n such that n leaves remainder rj on division by pj for each j.

Figure 15.5 Chongzhi

Zhu Chongzhi and his computation of π Zhu Chongzhi (429-500) was born in He Bei Province, and he was an astronomer, mathematician, and engineer. Several generations of his family studied astronomy and calender. had talent in science and mathematics at very early age. He collected together earlier astronomical writings, made own astronomical observations, and made a new calendar.

His most important work was to determine π accurate to 7 digits: 3.1415926, by using 355 Liu Hui’s algorithm applied to a 12288-polygon. He also proposed using 113 for a close 22 approximation value of π, and 7 for a rough approximation of π. The reason why did Zhu Chongzi success was that when he used the which is a device made of small

96 bamboo bars, he actually used the concept of “zero” to denote “place” (e.g., 20103). 1 This value of π by Zhu would remain the most accurate approximation of π available for the next 900 years.

His son, Zhu Hengzhi, was also a mathematician who did some important work.

Systems of linear equations The book Nine Chapters (i.e., Jiuzhang Shuashu) contained problems on systems of linear equations. For example, one problem is as follows:2 The price of 1 acre of good land is 300 pieces of gold; the price of 7 acres of bad land is 500. One has purchased altogether 100 acres; the price was 10,000 pieces of gold. How much good land was bought and how much bad?

Figure 15.6 Terra Cotta Warriors from the tomb complex of China’s First Emperor.

In today’s language, the problem is to solve the system of equations: ( x + y = 100, 500 (1) 300x + 7 y = 10, 000. Here is the Chinese rule for the solution: Suppose there are 20 acres of good land and 80 2 of bad. Then the surplus is 1714 7 . If there are 10 acres of good land and 90 of bad, the 1The written notation of zero appear later in India and was imported to Europe in the 16th century. 2Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, 2009, p. 210.

97 3 deficiency is 571 7 . Then, as the solution procedure by the Chinese author, multiply 20 by 3 2 2 571 7 , 10 by 1714 7 , add the products, and finally divide the sum by the sum of 1714 7 and 3 1 1 571 7 . Them the amount of good land is 12 2 , and hence the amount of bad land is 87 2 acres. From today’s point of view, since y = 100 − x, the second equation in (1) becomes 500 300x + (100 − x) = 10, 000. 7

500 In other words, we look for x such that f(x) = 10, 000, where f(x) = 300x + 7 (100 − x). Since the graph of this function is a straight line, by linearity, if we choose distinct x1, x2 with f(x1) − f(x) 6= 0 and f(x) − f(x2) 6= 0, we should have x − x x − x 1 = 2 f(x1) − f(x) f(x) − f(x2) so that x f(x) − f(x ) + x f(x ) − f(x) x = 1 2 2 1 . f(x1) − f(x2) 1 If we take x1 = 20 and x2 = 10, we get the above solution x = 12 2 . However, the authors did not explain how the algorithm was obtained. The algorithm turned up in the Islamic world and then in western Europe over a thousand year later.

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