History of Mathematics in Ancient China: Key Achievements

Introduction

In ancient China, the science of mathematics was subsumed under the larger practice of suan chu, or the "art of calculation." The Chinese are believed to be one of the first civilizations to develop and use the decimal numeral system. Their early mathematical studies influenced science among neighboring Asian countries and beyond.

This paper examines the history of mathematical knowledge in China. It looks at early Chinese achievements in the field of mathematics, including the decimal system, the calculation of pi, the use of counting aids, and the application of mathematical principles to everyday life. It also examines the influence of Indian and, later, European mathematical knowledge on Chinese mathematics.

Early China

Unlike the ancient Greeks, who prized knowledge for its own sake, much of the scientific study conducted in ancient China was spurred by practical everyday needs. Because of its geographic location, China was prone to devastating floods, particularly along the powerful Yangtze and Yellow Rivers. Every year, the banks would overflow, destroying crops and claiming lives.

Suan chu was thus developed to cover a wide array of practical and spiritual concerns. Subjects as diverse as religion and astronomy were drawn upon to devise ways to control the floods (Martzloff 21–22). Mathematics was an integral aspect of suan chu, particularly in the construction of dams strong enough to reinforce the river banks and in the development of the Chinese calendar to record and predict the monsoon season.

The use of decimal notation in China dates back to the Shang Dynasty, which lasted from 1700 to 1027 BC. Chinese legend holds that numbers were a divine gift from a river tortoise, conveyed through diagrams known as Lo shu. These diagrams were believed to contain the principles of Chinese mathematics, which were in turn rooted in the concepts of Yin and Yang — the complementary opposites. This cosmological framework was reflected in the distinction between even and odd numbers.

Similar to the decimal system used today, scholars of the Shang Dynasty employed traditional decimal notation. They had one symbol for each digit from 1 through 9 and additional symbols for 10, 100, 1,000, and 10,000. Thus, the Chinese numerical equivalent of the Western number 3,125 would be written as "3 times 1,000 plus 1 times 100 plus 2 times 10 plus 5."

By the Eastern Zhou Dynasty (770–221 BC), records show that Chinese mathematicians arranged their digits from left to right, similar to the present-day Arabic numeral system. They had a concept of "zero," using the digit 0 as a placeholder. To aid their calculations, mathematicians also used small bamboo counting rods, with a gap between the rods designating zero.

The following Han Dynasty saw important developments in Chinese mathematics. Around 100 BC, the publication of The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven by an unknown scholar demonstrated that the ancient Chinese held clear scientific theories regarding astronomy and circular orbits. The book uses calculations involving fractions, showing that Chinese scholars had a firm grasp of fractional numbers. It also applied the Pythagorean theorem to surveying and charting the heavens (Needham 44).

The most important book of ancient Chinese mathematics is the Nine Chapters on the Mathematical Art, written circa 100 BC to 50 AD. This volume collates much of Chinese mathematical knowledge from the beginning of recorded history to the start of the Han Dynasty.

The chapters address algorithms, the areas of plane figures, proportional distributions, the calculation of fair taxes, and methods for finding square and cube roots. They also cover the volumes of three-dimensional figures such as cubes, pyramids, and tetrahedrons, as well as further applications of the Pythagorean theorem (Swetz 12). The book additionally demonstrated how to solve quadratic equations using a modified square-root algorithm.

The Nine Chapters also discussed the use of negative numbers in solving linear equations — a significant development, since the practical orientation of earlier Chinese mathematics had not required the concept of negative or irrational numbers (Martzloff 52).

The renowned mathematicians of this period include Tsu Ch'ung Chi, whose approximation of pi was correct to six decimal places. His mathematical skills also helped him, as an astronomer, to calculate the solstice by measuring and recording the shadow of the sun.

Liu Hui (born 263 AD) wrote a commentary on the Nine Chapters. He further refined the calculation of pi by using polygons and employed sophisticated mathematical techniques to find the volume of a cylinder. Two centuries later, the mathematician and engineer Zu Chongzhi built on Liu's approximations to devise a more accurate formula for the volume of a sphere (Needham 48).

Despite these theoretical advances, much of ancient Chinese mathematics remained rooted in practical concerns. A large portion of the Nine Chapters, for example, consists of mathematical problems drawn from everyday life. Around 450 AD, Zhang Qiujian wrote a mathematical manual that included a solution to the famous "hundred fowls" problem:

"A cock is worth 5 qian, a hen 3 qian, and 3 chicks 1 qian; with 100 qian we buy 100 of them; how many cocks, hens, and chicks are there?"

Indian Influence

Qiujian's solution employed linear equations with three unknowns. Eliminating results that did not yield whole numbers, he found three valid combinations: 12 cocks, 4 hens, and 84 chicks; 8 cocks, 11 hens, and 81 chicks; or 4 cocks, 18 hens, and 78 chicks (Martzloff 16). The hundred fowls problem illustrates how even complex mathematical techniques were directed toward solving practical concerns.

By the 7th century, the Chinese had come into contact with Indian mathematicians and astronomers. An Indian astronomer translated important mathematical works into Chinese. One of these texts addressed the measurement of angles; another contained tables of calculated sine values for angles ranging from 0 to 90 degrees, given in 24-step increments.

These sine values helped Chinese mathematicians refine their astronomical calculations, enabling them to more accurately predict celestial events such as eclipses. These translations also introduced Chinese scholars to Hindu-Arabic notation, though these numerical symbols were not ultimately adopted (Martzloff 96–101).

Conclusion

The lack of any discernible influence today should not detract from the great achievements of ancient Chinese mathematics. Mathematical principles also underlay the development of more celebrated Chinese scientific innovations, such as gunpowder, paper money, and seismographs — the latter used to measure earthquakes as early as 1000 AD. It is through these scientific and technological developments that ancient Chinese mathematical principles continue to live on.