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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
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”
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 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.
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
multiplicative in nature.
additive
and
Zhoubi suanjing

Zhoubi Suanjing was 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.
 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.

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).
Jiuzhang Suanshu
The 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
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:

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.

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.

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.
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
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.
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.

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
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 Suanshu
Approximates 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

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.

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.

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.

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). Higherorder 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.