Development of Mathematics
in Ancient China

Mathematics knows no races or geographic boundaries;
for mathematics, the cultural world is one country.
~ Hilbert, David (1862-1943) ~

"Chinese mathematics," was defined by Chinese in ancient times as the "art of calculation" (suan chu). This art was both a practical and spiritual one, which covered a wide range of subjects from religion and astronomy to water control and administration.

The history of China is filled with many periods marked by flourishing culture and civilization, and others marked by competition and war between dynasties and invaders. By the year 1000 AD, the Chinese had invented seismographs to measure earthquakes. The Chinese also had ships big enough to sail around the world, but did not do so, and compasses with a magnetic needle used for navigation in 1119. China had developed many other inventions in science and technology, including gunpowder, paper money, moveable type and blast furnaces capable of producing cast iron. Among China's books were the New and More Detailed Pharmacopoeia of the Khai-Pao Reign Period.

The first true evidence of mathematical activity in China can be found in numeration symbols on tortoise shells and flat cattle bones (commonly called oracle bones), dated from the Shang dynasty (14th century B.C.).



These numerical inscriptions contain both tally and code symbols which are based on a decimal system, and they employed a positional value system. This proves that the Chinese were one of the first civilizations to understand and efficiently use a decimal numeration system. Early Chinese mathematics had a great influence on other later civilizations, in India, Japan, Korea and other counties.

Shang Numerals

The Shang numerals for the numbers one to nine were:



Indeed, the numeration system used in the modern world had its origins 34 centuries ago in Shang China. However, mathematics in China began much earlier in the development of the Chinese Calendar, flood-control measures, administration, and so on.

The need to control the flood-prone rivers of China, such as the Yangtze and Yellow Rivers, was an important factor in the development of mathematics in ancient China. The problem of providing a safe environment in a water-dependent society were solved using science and mathematics, including the construction of canals, dams, etc.

The Legend of Lo Shu

According to s legend, a mythological Emperor Yu received a divine gift from a Lo river tortoise. The gift was in the form of diagrams called Lo shu, which was believed to contain the principles of Chinese mathematics. One diagram, the magic square, was thought to possess magical qualities, and led to the development of the dualistic theory of Yin and Yang, where Yin represents even numbers (2, 4, 6, 8, 10...) and Yang represents odd numbers (1, 3, 5, 7, 9, 11...).



The Chinese also discovered the concept of zero and invented a wide variety of mechanical aids like counting boards besides writing numerous mathematics texts to assist them in mathematical calculation.


Chinese Counting Boards

By the time of the Han Dynasty (2nd Century B.C. to 4th Century A.D.), the Shang numerals were developed into a system and used on a counting board and a set of counting-rods (chousuan). Chinese computational methods were based on many mechanical aids, or early computers, including a large variety of counting boards. Indeed, the evolution and development of counting rod numerals continued for about 3000 years in China, from 14 century B.C. to 13th century A.D.



The abacus is perhaps the best known counting board, but it was only developed around the 15th century. Counting boards were divided into columns designating positional groupings of 10. Zero was indicated by a blank space on the counting board, and negative numbers were also used in mathematical calculations. Two types of rodswere used, red and black or positive and negative, representing Yin and Yang.

Before the dawn of the Christian era, over 2,000 years ago, the Chinese were solving systems of equations with a method similar to the modern transformations of matrices. Tangrams and other aids were also invented in China during the Chou dynasty. In addition, over the thousands of years of mathematical progress, the Chinese have invented a large variety, and countless numbers, of mathematical puzzles and games.

The linear units of measure were as follows:

1 chang = 10 ch'ih
1 pu (pace) = 5 ch'ih
1 ch'ih (foot) = 10 ts'un (inch)


Chinese Math Texts

The history of Chinese mathematics and mathematicians was mostly lost or destroyed over the centuries. For example, the despotic emperor Shih Huang-ti of the Ch'in dynasty (221-207 B.C.) ordered the burning of books in 213 B.C.. Scholars in the following Han period (206 B.C. to 220 A.D.) had to transcribe China's literary and scientifice traditions from memory or remaining fragments of scroll.



Knowledge of astronomy and other areas was often handed down from father to son, and only later recorded in texts. Unfortunately, very few texts dedicated to mathematical astronomy have survived.

Since the 16 century, Chinese math history has also been denied and ignored in the Western dominance of science and technology, both inside and outside China.

However, there are several existing Chinese applied mathematics texts, which are collections of problems and solutions organized in chapters according to their practical applications. These texts proves that the Chinese were the first society to use some of the most basic and advanced mathematical principles and concepts utilized in modern times. Two of these texts are the Chou Pei and Chiu Chang.

Chou Pei

It is oldest existing Chinese texts containing formal mathematical theories that were produced during the Han period. The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven (Chou Pei Suan Ching) is dated before the 3rd century B.C and contains various modern mathematical principles such as working with fractions using a common denominator, and proofs of many geometrical theories. The text contains an accurate process of division for finding out the square root of numbers.

In fact, the Chou Pei presents the oldest known proof of the right-angle triangle theory in the hsuan-thu diagram. This theory, commonly known as the "Pythagorean theorem," shows that the sum of the squares of the legs of a right triangle is equal to the squares of the hypotenuse or A2 + B2 = C2.



The Chou Pei was not an isolated academic text shared only by a few ancient Chinese mathematicians. The principles in the text were reflected in the popular approach known as chi-chu, or "the piling up of squares" which was a process of using geometry to solve algebraic problems.

Chiu Chang

Another 3rd century B.C. Han text, the Nine Chapters on the Mathematical Art (Chiu Chang Suan Shu), was very influential in Asian mathematics. This text was probably first written by Chang Tshang who made use of older works then in existence.

The Chiu Chang focuses on applied mathematics in engineering and administration and include nine distinct chapters on impartial taxation (chun shu), engineering works (shang kung), the surveying of land (fang thien), etc. In total, 246 problem situations are presented, from those involving the payment for livestock, weights and measures, currency and tax collection to the construction of canals and simultaneous linear equations (fang chheng).

Other Texts

Other important Chinese math texts include the Mathematical Classic of Sun Tzu (Sun Tzu Suan Ching) written in the 3rd century A.D. and The Ten Mathematical Manuals (Suanjing Shi Shu). The 13th century text, Detailed Analysis of the Mathematical Rules in the Nine Chapters (Hsiang Chieh Chiu Chang Suan Fa), proved the theory know as "Pascal's Triangle" 300 years before Pascal was born!