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 have influenced science among neighboring Asian countries and beyond.
This paper examines the history of mathematical knowledge in China. It looks at the early Chinese achievements in the field of mathematics, including the decimal system, 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 into Chinese mathematics.
Unlike the ancient Greeks who prized knowledge for its own sake, much of the scientific studies conducted in ancient China were 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 even taking lives.
Suan chu was thus developed, which covered a wide array of practical and spiritual concerns. Subjects as diverse as religion and astronomy were tapped to devise ways to control the floods (Martzloff 21-22). The science of mathematics was an integral aspect to the of suan chu, particularly in the construction of dams strong enough to shore up the river banks and in the development of the Chinese calendar to record and predict the monsoon season.
Though the worldwide use of the decimal notation system in China dates back to the Shang Dynasty, which lasted from 1700 to 1027 BC. Chinese legend states that the numbers were a divine gift from a river tortoise. These gifts were diagrams that were known as Lo shu. These diagrams were believed to contain the principles of Chinese mathematics, which in turn were rooted in the concepts of Yin and Yang, the complementary opposites. This was reflected in the concept of even and odd numbers.
Similar to the decimal system of today, the scholars of the Shang dynasty used traditional decimal notation. They had one symbol for the digits 1 through 9. In addition, however, they also had one symbol for the number 10, 100, 1000 and 10,000. Thus, the Chinese numerical equivalent of the Western number 3,125 would be written as "3 times 1000 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 space. To aid in their calculations, the mathematicians also used small bamboo counting rods. The placement of the rods designated their place in the decimal system, with a gap between the rods to designate zero.
The following Han Dynasty saw important developments in the field of Chinese mathematics. At circa 100 BC, the publication of The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven by an unknown Chinese scholar showed that the ancient Chinese had clear scientific theories regarding astronomy and the circular orbits. The book uses calculations involving fractions, showing that the Chinese then had a clear concept of fractional numbers. In addition, the book also used the Pythagorean theorem for surveying and charting the heavens (Needham 44).
The most important book of ancient Chinese mathematics is the Nine Chapters on the Mathematical Art, which was written circa 100 BC to 50 AD. This volume collates much of Chinese mathematical knowledge to from beginning to the start of the Han dynasty.
Chapters in this volume touch on algorhythms and the areas of plane figures, proportional distributions, the calculation of fair taxes and methods for calculating square and cube roots. It also had chapters on finding volumes of three-dimensional figures like cubes, pyramids and tetrahedrons and further applications of the Pythagorean theorem (Swetz 12). The book also showed how to solve quadratic equations by using a modified square root algorhythm.
The Nine Chapters also discussed the use of negative numbers in solving linear equations. This was an important development, since the practical orientation of earlier Chinese mathematics did not need the concept of negative or even irrational numbers (Martzloff 52).
The renowned mathematicians of this period include Tsu Ch'ung Chi, whose approximation of the value of pi was correct to six places. His mathematical skills helped Tsu, also 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 calculations of pi by using polygons and used sophisticated mathematical techniques to find the volume of a cylinder. Two hundred years later, a mathematician and engineer named Zu Chongzhi would build 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. Much of the Nine Volumes, for example, were a series of mathematical problems related to everyday life. At around 450 BC, Zhang Qiujian wrote a mathematical manual including a solution to the "hundred fowls" problem. This problem involved the following question:
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?"
Qiujian's solution involved linear equations containing three unknowns. Eliminating the results which did not involve whole numbers, Qiujian found three possible 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 is an example of how even complex mathematical techniques were geared to solving practical concerns.
By the 7th century, the Chinese had contact with Indian mathematicians and astronomers. An Indian astronomer translated important Indian mathematical works into Chinese. One of these books involved the measurement of angles. The other book contained tables of calculated sine values of angles that measured from 0 to 90 degrees. These values were given in 24 step increments.
These sine values helped Chinese mathematicians refine their astronomical calculations, helping them to better predict important celestial events like eclipses. These translations also introduced Chinese scholars to Hindu Arabic notation, though these numerical symbols were not adopted (Martzloff 96-101).
Tenth Century to Ming period
When Westerners think of Chinese mathematics, their thoughts invariably turn to the abacus. However, in light of the long history of mathematics in China, the abacus is a fairly recent invention, having been developed circa the 15th century. These counting boards were again based on the decimal system. A blank space on the counting board represented zero. Red and black rods represented the positive and negative numbers, in keeping with the opposite principles of Yin and Yang (Martzloff 10).
Prior to the abacus, however, mathematicians of this era were already involved in complex calculations. Qin Jiushao, who loved from 1202 to 1261, developed a method for finding linear congruences using cubic equations, a technique that involved Euclidean geometry.
Around a century earlier, Jia Xian had devised a way to extract square and cube roots from positive integers. He also employed the Pascal Triangle, centuries before French mathematician Blaise Pascal popularized the pyramid of numerical values (Martzloff 17). By the late 12th to the late 13th century, mathematician Li Ye was delving into the more abstract field of algebra, by applying algebraic solutions to geometric problems.
Unfortunately, Li Ye was the last important figure of ancient mathematics. The Mongol invasion during the Yuan Dynasty (1279 to 1368 AD) resulted in the destruction of many important old texts. The ones that survived were either destroyed or forbidden…