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.

Particularly important was mathematical astronomy which attracted attention from rulers w

ho had the royal observatory and employed mathematicians, astronomers, and astrologers.

Mathematicians were responsible for establishing the algorithms of the calendar-making sy

stems. So, mathematics served the needs of mathematical astronomy. Calendar-makers wer

e required a high degree of precision in prediction. They worked hard at improving numeric

al method, which was the principal method of Chinese calendar-making systems. It was val

ued for high accuracy in prediction and computation.

Some scholars think that Chinese mathematicians discovered the concept of zero, while oth

ers express the opinion that they borrowed it from the Hindus at the meeting place of the Hi

ndu and Chinese cultures in south-east Asia. The Chinese symbol for zero developed from t

he 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 h

at their “sunya”

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.

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."

Chinese NumeralsIn 1899 a major discovery was made at the archaeological site at the villa

ge 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 Chines

e characters. The site had been the capital of the kings of the Late Shang dynasty (t

his 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 di

scovered there had been used as part of religious ceremonies. Questions were inscri

bed on one side of a tortoise shell, the other side of the shell was then subjected to t

he heat of a fire, and the cracks which appeared were interpreted as the answers to t

he questions coming from ancient ancestors.

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.

Zhoubi suanjing Zhoubi Suanjing was essentially an astronomy text,

thought to have been compiled between 100 BC a

nd 100 AD, containing some important mathemati

cal sections. The book was listed as the first and o

ne of the most important of all the texts included i

n the Ten Mathematical Classics. The text measur

es the positions of the heavenly bodies using shad

ow gauges which are also called gnomons.

How a gnomon might be used is described i

n a conversation in the text: Duke of Zhu: How great is the art of numbers? Tell me something abou

t 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; whe

n 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.

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 Suanj

ing 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).

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.

Afterwards, the Jiuzhang Suanshu have been annotated by many mathematicians, the m

ost famous ones including Liu Hui (in 263AD) and Li Chunfeng (in 656AD). The editio

n published by the Northern Song government in 1084 was the earliest mathematical bo

ok in the world. The book was introduced to Korea and Japan during the Sui and Tang d

ynasties (581-907). Now, it has been translated into several languages, including Japanes

e, 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 a

nd procedures. Each chapter deals with specific field of questions. Here is a short descri

ption of each chapter:

Chapter 1, Field measurement(“Fang tian”): systematic discussion of algorithms usi

ng counting rods for common fractions for GCD, LCM; areas of plane figures, squa

re, rectangle, triangle, trapezoid, circle, circle segment, sphere segment, annulus. R

ules 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 a

bout 850 AD.

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, p

roblems 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 are

a 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 frustu

ms, 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.

Chapter 6, Fair taxes(“Jun shu”): discusses the problems in connection with the tim

e required for people to carry their grain contributions from their native towns to th

e 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 a

nd double false position to solve difficult problems.

Chapter 8, Rectangular arrays(“Fang cheng”): gives elimination algorithm for solvi

ng systems of three or more simultaneous linear equations. Introduces concept of p

ositive and negative numbers (red reds for positive numbers, black for negative nu

mbers). Rules for addition and subtraction of signed numbers.

Chapter 9, Right triangles(“Gou gu”): applications of Pythagorean theorem and sim

ilar triangles, solves quadratic equations with modification of square root algorith

m, only equations of the form x^2 + a x = b, with a and b positive.

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.

5. Introducing the concepts of positive and negative numbers, more than 600 years earlier th

an the West.

6. Developing a general solution formula for the Pythagorean problems (problems of Gou g

u), 300 years earlier than the West.

7. Putting forward theories of calculating areas and volumes of different shapes and figures.

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.

The Abacus

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).

Sun Zi (c. 250? C.E.) : Wrote his mathematical manual. Includes "Chinese rema

inder problem“ or “problem of the Master Sun”: find n so that upon division by 3 y

ou get a remainder of 2, upon division by 5 you get a remainder of 3, and upon divi

sion by 7 you get a remainder of 2. His solution: Take 140, 63, 30, add to get 233, s

ubtract 210 to get 23.

Liu Hui (c. 263 C.E.)

Commentary on the Jiuzhang Suanshu

Approximates pi by approximating circles polygons, doubling the number of si

des to get better approximations. From 96 and 192 sided polygons, he approxi

mates 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

Haidao suanjing (Sea Island Mathematical Manual). Originally appendix to co

mmentary on Chapter 9 of the Jiuzhang Suanshu. Includes nine surveying prob

lems 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 equati

ons in three unknowns, the "hundred fowls problem"

Zu Chongzhi (429-500): Astronomer, mathematician, engineer.

Collected together earlier astronomical writings. Made own astronomical obser

vations. Recommended new calendar.

Determined pi to 7 digits: 3.1415926. Recommended use 355/113 for close app

rox. and 22/7 for rough approx.

With father carried out Liu Hui's suggestion for volume of sphere to get accura

te formula for volume of a sphere.

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. Solve

d 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 Ob

servatory, translated two more texts, one of which described angle measurement (3

60 degrees) and a table of sines for angles from 0 to 90 degrees in 24 steps (3 3/4 d

egree) increments.

Hindu decimal numerals also introduced, but not adopted.

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 Ni

ne Sections), 81 problems of applied math similar to the Nine Chapters. Solution of

some higher-degree (up to 10th) equations. Systematic treatment of indeterminate s

imultaneous linear congruences (Chinese remainder theorem). Euclidean algorithm

for GCD.

Li Chih (a.k.a. Li Yeh) (1192-1279): Ceyuan haijing (Sea Mirror of Circle M

easurements), 12 chapters, 170 problems on right triangles and circles inscribed wit

hin or circumscribed about them. Yigu yanduan (New Steps in Computation), geom

etric problems solved by algebra.

Yang Hui (fl. c. 1261-1275): Wrote sevral books. Explains Jiu Xian's methods f

or solving higher-degree root extractions. Magic squares of order up through 10.

Guo Shoujing (1231-1316): Shou shi li (Works and Days Calendar). Higher-or

der differences (i.e., higher-order interpolation).

Zhu Shijie (fl. 1280-1303): Suan xue qi meng (Introduction to Mathematical St

udies), and Siyuan yujian (Precious Mirror of the Four Elements). Solves some hig

her degree polynomial equations in several unknowns. Sums some finite series incl

uding (1) the sum of n^2 and (2) the sum of n(n+1)(n+2)/6. Discusses binomial coe

fficients. Uses zero digit.