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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.