Chinese Mathematics Chinese Contributions to Modern Mathematics Early

Chinese Mathematics Chinese Contributions to Modern Mathematics

Early Chinese Civilizations • 1200 B. C - 300 B. C: Zhou Bi Suan Jing • Some of the oldest mathematical works known anywhere come from the Zhou Dynasty • Warring States Period (476 – 221 BC) • Ended the Zhou dynasty • 231 Qin Shu Huang (Qin Tyrant) ordered the destruction of books

Earliest Records • The Book of Master MO • Written under the Qin Tyrant despite the order to stop learning • Contained geometric results including proofs; but not quite the high level work that the Elements were.

Zhou Bi Suan Jing (周髀算經) • One of two major works from the Han Dynasty • • In the form of dialogue Astronomical calculations Geometry Introduction to properties of right triangles • Fractions

The Nine Chapters – Jiuzhang suanshu • Most influential of all Chinese mathematical books • Contains 246 problems • • • Surveying Agriculture Engineering Economics Solutions to Equations

Jiuzhang Suanshu The Nine Chapters of Mathematical Art contains 246 practical problems that might arise in the average day. This was used as the mathematical textbook of the age. What other civilization(s) did we study that taught mathematics in this way. What ancient artifact(s) gave us evidence for this historical conclusion? • Uganda, Ishango Bone • Babylonia, Plimpton Tablets • Egypt, Rhind Papyrus • Greece, Euclid’s Elements

A short summary of each of the nine chapters Based on commentaries by Liu Hui

Preamble • This common preamble to the Nine Chapters describes an arithmetic based on a base 10, presumably positional, number system. • It hints at the Chinese Abaci, but we are not entirely sure

Politically Charged Question • Did the Chinese develop a base ten positional number system and arithmetic based on it before the Indians who currently receive credit for this?

Best Current Answer • Maybe.

Best Current Explanation • Good chance the Chinese had it first and the Indians got the idea from them, if you only consider the number system being used. • If you consider a written number system, then the Indians are undoubtedly keeping the credit. • We will talk more when we do India

Chapter 1: Land Surveying • • • 38 problems Area problems: Triangles, Quadrilaterals and Circles Rules for arithmetic of fractions Accurate approximation of Euclidean Algorithm

$Chapter 1 Problems Problem 5: Now given a fraction 12/18. Tell: reducing it (to$

Chapter 1 Problems Problem 5: Now given a fraction 12/18. Tell: reducing it (to its lowest terms), what is obtained? Problem 6: Given another fraction 49/91. Tell: reducing it, what is obtained? The Rule for Reduction of Fractions If (the denominator and numerator) can be halved, halve them. If not, lay down the denominator and numerator, subtract the smaller number from the greater. Repeat this process to obtain GCD. Reduce them by the GCD.

Euclidean Algorithm Euclidean algorithm For any two numbers, m and n , if n < m, then we can arrive at the greatest common factor of n and m, gcd(n , m), by reiterating the process below Then, , the last nonzero remainder is the greatest common divisor.

Example

Chapter 2: Millet and Rice • • • 46 problems Exchange of goods Rate change Proportions Percentages A circular road around a hill is 325 li long. Three persons A, B, and C run along the road. A runs 150 li per day, B runs 120 li per day, and C runs 90 li per day. If they start at the same time from the same place, after how many days will they meet again?

Chapter 3: Distribution by Proportion • • • 20 problems Proportion Inverse Variation Compound Proportion Arithmetic Progression Geometric Progression

Chapter 4: Short Width • 24 problems • Area of rectangular field constant • Width of field increased • What is length of field?

Chapter 5: Civil Engineering • 28 problems • Constructions of canals, ditches, dykes • Volumes of Solids • Method of Exhaustion

Chapter 6: Fair Distribution of Goods • • • 28 problems A cistern is filled through five canals. Open Ratio and proportion the first canal and the cistern fills in 1/3 day; with the second, it fills in 1 day, with the third, Travelling in 2 ½ days; with the fourth in 3 days, and with the fifth, in 5 days. If all canals are Taxation opened, how long will it take to fill the Sharing cistern?

Chapter 7: Excess and Deficit • 20 problems • Rule of False Position • Linear Equations Certain Items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.