In ancient China, mathematics was already one of the most advanced basic sciences. The development of traditional mathematics in China can be divided into several periods. Its formative period was from ancient times to the Spring and Autumn era (770–476 BCE). A rudimentary foundation was laid down between the Warring States period (475–221 BCE) and the beginning of the Tang dynasty (early seventh century). From the middle of the Tang dynasty to the middle of the Yuan dynasty (mid-seventh–mid-fourteenth centuries), it reached a peak but then it declined and continued to do so until the middle of the Ming dynasty (ca. late fifteenth century) when the influence of Western science can be seen. Mathematics in China was closely associated with practical application and an emphasis was placed on computation. The procedural and mechanical skills developed for computation were exceptional.

A long time ago, we can surmise that the ancestors of the Chinese people learned about numbers and shapes from nature. Among the archaeological artifacts excavated from the Neolithic, we can find ceramic objects bearing geometric shapes as well as codes representing numbers. In the beginning, people counted simply: one man, two men, one sheep, two sheep, etc. We know from contact with some tribal villages not too long ago, that people were only able to count up to five; any number after which was considered “many.” When a number was used, such as number 5, to mean five people, five sheep, or five of something else, an era of abstract numerical concepts began. With the help of this concept, codes representing numbers started to take form.

In addition to numbers, people also invented the compass, the carpenter’s square, and surveying instruments. The decimal system and positional notation can already be seen on oracle bones. At the end of the Yin and beginning of the Zhou dynasty (ca. early twelfth century), rudimentary knowledge of what, in the West, is called the Pythagorean theorem existed. When the Duke of Zhou designed the rites and rituals, at the beginning of the Zhou dynasty, mathematics was one of six major topics in the education of aristocratic scions. No later than the Spring and Autumn period were people able to proficiently master the very advanced decimal system, positional notation, the multiplication table, the elementary arithmetic of natural numbers, and fractions.

In the Warring States period, numerous specialists put forward their own ideas about numbers and eventually the discipline of mathematics evolved into nine branches (jiushu): fangtian (surveying of land), sumi (percentages and proportions relating to grains ‘millet and rice’), chafen (distribution by progression), shaoguang (calculation of the area and dimensions of farmland), shanggong (diminishing breath), junshu (impartial taxation), yingbuzu (excess and deficiency), fangcheng (the way of calculation by tabulation), and pangyao or gougu (base and altitude). Together they form the framework of traditional Chinese mathematics. Many mathematical books were written during this period, such as the Suanshu shu (Book of numbers and computation), Zhou Bi suan jing (Mathematical classic of Zhou Bi), and Jiuzhang suanshu (Nine chapters on the mathematical art). They centered the study of mathematics on abstract calculation, solution-oriented examples, and close association with practical experience. From here onward, the future characteristics and style of Chinese and Asian mathematics of the next two thousand years were anchored.

The Jiuzhang suanshu zhu (Commentary on the nine chapters on the mathematical art), written by Liu Hui (fl. 263) in the Wei dynasty, consolidated and fleshed out many important theorems such as how to find the area or volume of a geometric form after cutting it into pieces, a study of cross-sections, a theory of equalization, and the ratio of a circle’s circumference to its diameter. In proving all the theorems by logical deductive reasoning Liu Hui laid down the theoretical foundation of traditional Chinese mathematics. At the peak of the Tang dynasty, China witnessed great advances in productivity and social improvement. In 1084 during the Northern Song dynasty, the Palace Library printed nine mathematical texts form the Han and Tang dynasties, which were the first printed mathematical works in the world. With works such as Suanxue qimeng (Primer on mathematical studies) by Zhu Shijie (fl. 1299) a high water mark is reached in the development of rod calculus, and the research of shortcuts in multiplication and division, laying down the groundwork for the abacus and created a hot bed for abacus skills to flourish. Great achievements were also made in formulas for finding the roots of high degree polynomials, congruence of the first degree, method of the heavenly elements or the method of the celestial unknown, finding the roots of multiple-variable high degree polynomials, etc.

Although mathematics in the Ming dynasty developed slower than it had in the Yuan dynasty, the popularity of the abacus reached an all-time high as it replaced rods as the go-to tool of calculation. Suanfa tongzong (Systematic treatise on arithmetic) by Cheng Dawei (fl. 1593) helped make the abacus popular and was influential in Korea, Japan, and Southeast Asia. At the end of the Ming dynasty, Jesuit missionaries such as Matteo Ricci (1552–1660) helped to spread basic Western mathematics to China, including geometry, algebra, and trigonometry. The Chinese translation of Euclid’s Elements, a joint effort by Ricci and Xu Guangqi (1562–1633), marked the first step in the Sino-Western mathematical exchange.