Characteristics of Chinese mathematics

1. Characteristics of Chinese mathematics • Chinese mathematics is characterized by a practical tradition. Many scholars held that practical appliance prevented Chinese mathematics from developing into modern science like Greece mathematics that is characterized by a theoretical tradition. From the historical perspective, Chinese mathematics served the needs of the society that was geographically isolated from the outer world. The Chinese needed controlling the flood prone Yangtze and Yellow Rivers. Mathematics helped solve the problem of a safe environment in a water-dependent society.

2. Particularly important was mathematical astronomy which attracted attention from rulers who had the royal observatory and employed mathematicians, astronomers, and astrologers. Mathematicians were responsible for establishing the algorithms of the calendar-making systems. So, mathematics served the needs of mathematical astronomy. Calendar-makers were required a high degree of precision in prediction. They worked hard at improving numerical method, which was the principal method of Chinese calendar-making systems. It was valued for high accuracy in prediction and computation. • Some scholars think that Chinese mathematicians discovered the concept of zero, while others express the opinion that they borrowed it from the Hindus at the meeting place of the Hindu and Chinese cultures in south-east Asia. The Chinese symbol for zero developed from the circle to denote the empty space in a number. Although it is generally accepted that zero was first used by the Hindu, the Chinese had “ling” (= “nothing”) long before the Hindus hat their “sunya”

3. A brief outline of the history of Chinese mathematics • Numerical notation, arithmetical computations, counting rods • Traditional decimal notation -- one symbol for each of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000, and 10000. Ex. 2034 would be written with symbols for 2, 1000, 3, 10, 4, meaning 2 times 1000, plus 3 times 10, plus 4. • Calculations performed using small bamboo counting rods. The positions of the rods gave a decimal place-value system, also written for long-term records. 0 digit was a space. Arranged left to right like Arabic numerals. Back to 400 B.C.E. or earlier.

4. Addition: the counting rods for the two numbers placed down, one number above the other. The digits added (merged) left to right with carries where needed. Subtraction similar. • Multiplication: multiplication table to 9 times 9 memorized. Long multiplication similar to ours with advantages due to physical rods. Long division analogous to current algorithms, but closer to "galley method."

5. Chinese Numerals In 1899 a major discovery was made at the archaeological site at the village of Xiao dun in the An-yang district of Henan province. Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese characters. The site had been the capital of the kings of the Late Shang dynasty (this Late Shang is also called the Yin) from the 14th century BC. The last twelve of the Shang kings ruled here until about 1045 BC and the bones and tortoise shells discovered there had been used as part of religious ceremonies. Questions were inscribed on one side of a tortoise shell, the other side of the shell was then subjected to the heat of a fire, and the cracks which appeared were interpreted as the answers to the questions coming from ancient ancestors.

6. The importance of these finds, as far as learning about the ancient Chinese number system, was that many of the inscriptions contained numerical information about men lost in battle, prisoners taken in battle, the number of sacrifices made, the number of animals killed on hunts, the number of days or months, etc. The number system which was used to express this numerical information was based on the decimal system and was both additive and multiplicative in nature.

7. Zhoubi suanjing • Zhoubi Suanjingwas essentially an astronomy text, thought to have been compiled between 100 BC and 100 AD, containing some important mathematical sections. The book was listed as the first and one of the most important of all the texts included in the Ten Mathematical Classics. The text measures the positions of the heavenly bodies using shadow gauges which are also called gnomons.

8. How a gnomon might be used is described in a conversation in the text: • Duke of Zhu: How great is the art of numbers? Tell me something about the application of the gnomon. • Shang Gao: Level up one leg of the gnomon and use the other leg as a plumb line. When the gnomon is turned up, it can measure height; when it is turned over, it can measure depth and when it lies horizontally it can measure distance. Revolve the gnomon about its vertex and it can draw a circle; combine two gnomons and they form a square.

9. Zhoubi Suanjing contains calculations of the movement of the sun through the year as well as observations of the moon and stars, particularly the pole star. • Perhaps the most important mathematics which is included in the Zhoubi Suanjing is related to the Gougu rule, which is the Chinese version of the Pythagoras Theorem. • The big square has area (a+b)^2 = a^2 +2ab + b^2. The four "corner" triangles each have area ab/2 giving a total area of 2ab for the four added together. Hence the inside square (whose vertices are on the outside square) has area (a^2 +2ab + b^2) - 2ab = a^2 + b^2. Its side therefore has length ( a^2 + b^2). Therefore the hypotenuse of the right angled triangle with sides of length a and b has length ( a^2 + b^2).

10. Jiuzhang SuanshuThe Nine Chapters on the Mathematical Art • This book is the most influential of all Chinese mathematical works in the history of Chinese mathematics. It is the longest surviving and one of the most important in the ten ancient Chinese mathematical books. The book was co-compiled by several people and finished in the early Eastern Han Dynasty (about 1st century), indicating the formation of ancient Chinese mathematical system. It became the criterion of mathematical learning and research for mathematicians of later generations ever since then.

11. Afterwards, the Jiuzhang Suanshu have been annotated by many mathematicians, the most famous ones including Liu Hui (in 263AD) and Li Chunfeng (in 656AD). The edition published by the Northern Song government in 1084 was the earliest mathematical book in the world. The book was introduced to Korea and Japan during the Sui and Tang dynasties (581-907). Now, it has been translated into several languages, including Japanese, Russian, German, English and French, and become the basis for modern mathematics. • The book is broken up into nine chapters containing 246 questions with their solutions and procedures. Each chapter deals with specific field of questions. Here is a short description of each chapter:

12. Chapter 1, Field measurement(“Fang tian”): systematic discussion of algorithms using counting rods for common fractions for GCD, LCM; areas of plane figures, square, rectangle, triangle, trapezoid, circle, circle segment, sphere segment, annulus. Rules are given for the addition, subtraction, multiplication and division of fractions, as well as for their reduction. Also, rules are given for the segment of a circle as A = 1/2 (c + s) s , where A is the area, c the chord & s the sagitta of the segment. The same expression is found in the works of the Indian mathematician Mahavira about 850 AD.

13. Chapter 2, Cereals(“Sumi”): deals with percentages and proportions. It reflects the management and production of various types of grains in Han China. • Chapter 3, Distribution by proportion(“Cui fen”): discusses partnership problems, problems in taxation of goods of different qualities, and arithmetical and geometrical progressions solved by proportion. • Chapter 4, What width?(“Shao guang”): finds the length of a side when given th area or volume. Describes usual algorithms for square and cube roots. • Chapter 5, Construction consultations(“Shang gong”): concerns with calculation for constructions of solid figures such as cube, rectangular parallelepiped, prism frustums, pyramid, triangular pyramid, tetrahedron, cylinder, cone, prism, pyramid, cone, frustum of a cone, cylinder, wedge, tetrahedron, and some others. It gives problems concerning the volumes of city-walls, dykes, canals, etc.

14. Chapter 6, Fair taxes(“Jun shu”): discusses the problems in connection with the time required for people to carry their grain contributions from their native towns to the capital. There are also problems of ratios in connection with the allocation of tax burdens according to population. • Chapter 7, Excess and deficiency(“Ying bu zu”): uses of method of false position and double false position to solve difficult problems. • Chapter 8, Rectangular arrays(“Fang cheng”): gives elimination algorithm for solving systems of three or more simultaneous linear equations. Introduces concept of positive and negative numbers (red reds for positive numbers, black for negative numbers). Rules for addition and subtraction of signed numbers. • Chapter 9, Right triangles(“Gou gu”): applications of Pythagorean theorem and similar triangles, solves quadratic equations with modification of square root algorithm, only equations of the form x^2 + a x = b, with a and b positive.

15. The book's major achievements: 1. Devising a systematic treatment of arithmetic operations with fractions, 1,400 years earlier than the Europeans. 2. Dealing with various types of problems on proportions, 1,400 years earlier than the Europeans. 3. Devising methods for extracting square root and cubic root, which is quite similar to today's method, several hundred years earlier than the Western mathematicians. 4. Developing solutions for a system of linear equations, about 1,600 years earlier than the Western mathematicians.

16. 5. Introducing the concepts of positive and negative numbers, more than 600 years earlier than the West. 6. Developing a general solution formula for the Pythagorean problems (problems of Gou gu), 300 years earlier than the West. 7. Putting forward theories of calculating areas and volumes of different shapes and figures.

17. The Abacus • In about the fourteenth century AD the abacus came into use in China. Certainly this, like the counting board, seems to have been a Chinese invention. In many ways it was similar to the counting board, except instead of using rods to represent numbers, they were represented by beads sliding on a wire. Arithmetical rules for the abacus were analogous to those of the counting board (even square roots and cube roots of numbers could be calculated) but it appears that the abacus was used almost exclusively by merchants who only used the operations of addition and subtraction.

18. Here is an illustration of an abacus showing the number 46802. For numbers up to 4 slide the required number of beads in the lower part up to the middle bar. For example on the right most wire two is represented. For five or above, slide one bead above the middle bar down (representing 5), and 1, 2, 3 or 4 beads up to the middle bar for the numbers 6, 7, 8, or 9 respectively. For example on the wire three from the right hand side the number 8 is represented (5 for the bead above, three beads below).

19. Sun Zi (c. 250? C.E.) : Wrote his mathematical manual. Includes "Chinese remainder problem“ or “problem of the Master Sun”: find n so that upon division by 3 you get a remainder of 2, upon division by 5 you get a remainder of 3, and upon division by 7 you get a remainder of 2. His solution: Take 140, 63, 30, add to get 233, subtract 210 to get 23. • Liu Hui (c. 263 C.E.) • Commentary on the Jiuzhang SuanshuApproximates pi by approximating circles polygons, doubling the number of sides to get better approximations. From 96 and 192 sided polygons, he approximates pi as 3.141014 and suggested 3.14 as a practical approximation. States principle of exhaustion for circles Suggests Calvalieri's principle to find accurate volume of cylinder

20. Haidao suanjing (Sea Island Mathematical Manual). Originally appendix to commentary on Chapter 9 of the Jiuzhang Suanshu. Includes nine surveying problems involving indirect observations. • Zhang Qiujian (c. 450?):Wrote his mathematical manual. Includes formula for summing an arithmetic sequence. Also an undetermined system of two linear equations in three unknowns, the "hundred fowls problem" • Zu Chongzhi (429-500):Astronomer, mathematician, engineer. • Collected together earlier astronomical writings. Made own astronomical observations. Recommended new calendar. • Determined pi to 7 digits: 3.1415926. Recommended use 355/113 for close approx. and 22/7 for rough approx. • With father carried out Liu Hui's suggestion for volume of sphere to get accurate formula for volume of a sphere.

21. Liu Zhuo (544-610):Astronomer Introduced quadratic interpolation (second order difference method). • Wang Xiaotong (fl. 625):Mathematician and astronomer. Wrote Xugu suanjing (Continuation of Ancient Mathematics) of 22 problems. Solved cubic equations by generalization of algorithm for cube root. • Translations of Indian mathematical works.By 600 C.E., 3 works, since lost. Levensita, Indian astronomer working at State Observatory, translated two more texts, one of which described angle measurement (360 degrees) and a table of sines for angles from 0 to 90 degrees in 24 steps (3 3/4 degree) increments. • Hindu decimal numerals also introduced, but not adopted.

22. Yi Xing (683-727) tangent table. • Jia Xian (c. 1050): Written work lost. Streamlined extraction of square and cube roots, extended method to higher-degree roots using binomial coefficients. • Qin Jiushao (c. 1202 - c. 1261):Shiushu jiuzhang (Mathemtaical Treatise in Nine Sections), 81 problems of applied math similar to the Nine Chapters. Solution of some higher-degree (up to 10th) equations. Systematic treatment of indeterminate simultaneous linear congruences (Chinese remainder theorem). Euclidean algorithm for GCD. • Li Chih (a.k.a. Li Yeh) (1192-1279):Ceyuan haijing (Sea Mirror of Circle Measurements), 12 chapters, 170 problems on right triangles and circles inscribed within or circumscribed about them. Yigu yanduan (New Steps in Computation), geometric problems solved by algebra.

23. Yang Hui (fl. c. 1261-1275):Wrote sevral books. Explains Jiu Xian's methods for solving higher-degree root extractions. Magic squares of order up through 10. • Guo Shoujing (1231-1316):Shou shi li (Works and Days Calendar). Higher-order differences (i.e., higher-order interpolation). • Zhu Shijie (fl. 1280-1303):Suan xue qi meng (Introduction to Mathematical Studies), and Siyuan yujian (Precious Mirror of the Four Elements). Solves some higher degree polynomial equations in several unknowns. Sums some finite series including (1) the sum of n^2 and (2) the sum of n(n+1)(n+2)/6. Discusses binomial coefficients. Uses zero digit.