In the colorful treasure trove of Chinese history and culture, mathematics (math) is undoubtedly a resplendent and glaring pearl. Boasting paramount values and a unique position in the world civilization history, math is another important contribution China has made to the world besides the Four Great Ancient Inventions of papermaking,gunpowder,compass, andprinting.

The math achievements in ancient China boasted both a systematic theory and abundant inventions. Up to the 16th century, China was still leading the world in many aspects of math.

With a time-honored tradition, China maintained the longest period of development among the Four Great Ancient Civilizations of ancient China, Egypt, Babylon, and India, experiencing three major development climaxes in theHan Dynasty(206BC-220AD), the Wei, Jin, Northern and South dynasties period (220-581), and the Song and Yuan dynasties (960-1368). The development process can be generally classified into the following periods:

 The burgeoning period prior to theQin Dynasty(221-207BC)

The drainage areas of the Huanghe and Yangtze rivers have always been the cradles of Chinese culture. In the inscriptions on tortoise shells or animal bones of theShang Dynasty(16th-11th century BC) unearthed inYin ruins(ruins of the capital city of the late Shang Dynasty near Xiaotun Village,AnyangCity of Central China'sHenan Province), characters denoting numbers abound. There are altogether 13 symbols representing the numbers from 1 to 10, 100, 1,000, and 10,000, with the biggest number found on the oracle bones being 30,000.

Suan chou, literally meaning "counting rods," is an ancient Chinese calculating tool. When the suan chou first appeared cannot be determined now, but archeological finds show that the calculating system was already popular by theSpring and Autumn Period(770-476BC). The suan chou system built a good foundation for the four fundamental operations of arithmetic before being gradually replaced by theabacusat the end of theYuan Dynasty(1271-1368). It is fair to say that the great mansion of ancient Chinese math was built on the foundation of thesuan chou.

The achievements in geometry were also prominent. According to the Records of the Great Historian by Sima Qian, Emperor Yu of theXia Dynasty(21st-16th century BC) used various construction and measurement tools like drawing compasses, rulers, and ropes at the same time that the Gougu theorem (Chinese name for the Pythagorean theorem) was discovered.

In theWarring States Period(770-476BC), the Book of Diverse Crafts, known as Kao Gong Ji in Chinese, was written. The book summarizes the standards of handicrafts technology at the time, includes some contents about measurement, and contains some geometric knowledge like the concept of angles.

The contention of numerous schools of thought in the Warring States Period also promoted the development of math, with some schools summarizing and epitomizing many abstract notions in math. For instance,Mo Jing (Technological Volumes of Mozi)offers some famous propositions like "a circle is formed with one center and semi diameter of the length," as well as the definitions of finite and infinite. However, although the definitions of many geometric concepts and thought of the limit are all very valuable mathematic thoughts, this trend of emphasis on abstractness and strict logics were not very well inherited and developed.

In addition,Yijing (the Book of Changes)boasts the germination of combinatorics (math involving the combination of items), and reflects the thought of the binary system.

 Primary stage from the Han to the Tang dynasties (206BC-907AD)

This period spans many dynasties of more than 1,000 years. Monographs in systematizing and theorizing abundant mathematic knowledge gradually appeared in the Han Dynasty.

Zhoubi Suanjing is a classic work on astronomy and math written in the first century BC during theWestern Han Dynasty(206BC-24AD).

The Jiuzhang Suanshu (Nine Chapters on the Mathematical Art)is the longest surviving and one of the most important of the ten ancient Chinese math books. The book was co-compiled by several people and finished in the earlyEastern Han Dynasty(25-220), in about the 1st century, indicating the formation of the ancient Chinese math system. It became the criterion of mathematical learning and research for mathematicians of later generations.

Chinese math experienced substantial development theoretically in the Wei and Jin dynasties (220-420). The works done by Zhao Shuang and Liu Hui are generally considered as the beginning of the ancient Chinese math theory system.

Zhao Shuang was the earliest Chinese mathematician to prove math theorems and formulas, and he also made detailed notes to the classic work Zhoubi Suanjing.

Liu Hui noted the other mathematic canon,Jiuzhang Suanshu. His notes were not only a general explanation and induction, but also contained many innovations by way of introducing the book. Besides, he also wrote his own bookHaidao Suanjing, which focuses on solving some problems in measurement. Another important contribution by Liu was his invention of cyclotomy (dividing a circle into equal parts), which provided a theoretic foundation and scientific method for the research of the pi (the ratio of a circle's circumference to its diameter).

Despite longtime social unrest and division in theNorthern and Southern dynasties(420-581), mathematics still flourished vigorously. Many professional works, represented by Sunzi Suanjing,Xiahouyang Suanjing,and Zhang Qiujian Suanjian, were produced during the period.

The father and son of Zu Chongzhiand Zu Rihuan significantly advanced the development of traditional math on the basis of Jiuzhang Suanshunoted by Liu Hui, via an emphasis on mathematical thoughts and reasoning. In addition, they also made prominent contributions in astronomy. Unfortunately, their work,Zhuishu was lost.

The large-scale constructions and buildings in the Sui Dynasty(581-618) promoted the development of mathematics as a secondary effect.Jigu Suanjing, composed by Wang Xiaotong at the beginning of theTang Dynasty(618-907) mainly discussed some mathematic problems that appeared in civil engineering.

The math education in the Tang Dynasty advanced rapidly. Guozijian, also known as theImperial College, set up its math department in 656. The textbooks for the students in the math department were Suanjing Shishu, meaning ten important books in mathematics, compiled and noted by Li Chunfeng and his coworkers. The Suanjing Shishu textbooks were critical for the preservation of ancient Chinese mathematical classics.

Quadratic interpolation (number root-finding) method was created to meet the need of a calendar, laying a solid foundation for its latter development from the Song to the Yuan dynasties (960-1368). In the late Tang Dynasty, the calculating skills were further improved and popularized.

 The zenith age from the Song to the Yuan dynasties

After the demise of the Tang Dynasty, the following Five Dynasties(907-960) were preoccupied with chaotic tangled warfare. When the Song Dynasty (960-1279) finally reunited China in 960, agriculture, the handicraft industry, and business flourished immediately, and science and technology also advanced rapidly. The suan chou method reached the full bloom from the 11th to 14th century, in which Chinese math enjoyed unprecedented development with abundant attainments. A batch of famous mathematicians and written works abounded during the period.

The most famous works included:Huangdi Jiuzhang Suanfa Xicaoby Jia Xian in the middle of the 11th century;Yigu Genyuanby Liu Yi in the middle of the 12th century;Shushu Jiuzhang by Qin Jiushao in 1247;Ceyuan Haijing in 1248 and Yigu Yanduan in 1259 by Li Ye;Xiang Jie Jiuzhang Suanfa in 1261,Riyong Suanfa in 1262, and Yang Hui Suanfa in 1274-1275 by Yang Hui; and Suanxue Qimeng in 1299 and Siyuan Yujian in 1303 by Zhu Shijie.

Math in the Song and Yuan dynasties reached the zenith of ancient Chinese math and even the contemporary world math in many aspects. For instance, one major achievement was the solutions to the equation of higher degrees, in which symbols were introduced for the first time in Chinese math. The other attainments included the new development of the Gougu Theorem, the use of decimal fractions, and the appearance of the abacus, and so on.

During this period, the folk math education had some development, and the communication between China and the Islamic countries in math also began.

 The period when Western math passes in 

This period is about 500 years long from the mid-14th century when theMing Dynasty(1368-1644) was set up to the beginning of the 20th century when theQing Dynasty(1644-1911) demised. With the exception of the development of calculating with the abacus, math experienced acomprehensive decline. The possible reasons may include the limitations of the abacus, the deletion of math content in the 13th century examination system, and the rise of the BaguImperial Examinationsystem in the Ming Dynasty; many math historians are still discussing other possible reasons.

At the end of the 16th century, Western elementary math began to be passed to China, propelling the amalgamation of Chinese and Western math sciences in China. After theOpium Warin 1840, higher mathematics from the West began to spread to China, and Chinese math began to enter a period of mainly learning from the West. It was only in the 19th century that China really began to research contemporary algebra.

The most prominent achievement in the Ming Dynasty was the popularization of the abacus. Books about abacus calculations abounded during the time, and the abacus theoretic system was formed when Cheng Dawei completed Zhizhi Suanfa Tongzong in 1592. With the prevalence of the abacus, thesuan choualmost became extinct, which in turn led to the gradual disappearance of ancient math based on the suan chou. Math as a science became stagnant for a long time.

Although earlier, Indian mathematical and astronomical knowledge had passed to China during the Sui and the beginning of the Tang dynasties, the influence was very limited. Near the end of the 16th century, Western missionaries began to

come to China, translating a number of Western mathematical monographs with Chinese scholars. The first of these, and also one of the most important, was the Chinese edition of Euclid's Elements interpreted by Xu Guangqi and Matteo Ricci in 1607. The strict logic system and deduction method in Euclid's work were highly valued by Xu, who applied the same methods in the books that he wrote. Besides, most of the nouns in the translated version of Euclid's Elementswere for the first time created, and are still in use today. 

Among the imported knowledge during the time, the second most important, only after geometry, was trigonometry. Previously, the knowledge on trigonometry was scarce, but it developed rapidly afterwards.

In the Qing Dynasty, the outstanding representative mathematician achieving a comprehensive and thorough understanding of both Chinese and Western math was Mei Wending, who believed in traditional Chinese math. He made thorough researches in ancient math but could also correctly make sense of Western math, helping its growth in China. EmperorKangxi, with his fervent penchant in science and technology, ordered the compilation of Shuli Jingyun, which was a comprehensive and primary math book, in 1723.

After the Opium War, the policy of exclusion was forcibly abandoned, hence contributing the second wave of math work translations. Many math glossaries were created at the time, and are still being used.

 The period of contemporary and modern development

The period spans from the beginning of the 20th century to the present, and is usually divided into stages with the founding of thePeople's Republic of Chinain 1949 as the watershed. The contemporary mathematics began with students studying abroad at the end of the Qing Dynasty and the beginning ofthe Republic of China(1911-1949), many of whom became famous mathematicians or math educators after they came back to China. They also made great contributions to the math education field at universities across the country.

In 1936, the China Mathematics Academy Journal and Mathematics Magazine appeared, symbolizing a further development in modern math researches.

More than 600 kinds of monographs were published before the founding of the People's Republic of China, and Chinese mathematicians attained many achievements in a multiple of areas.

The Chinese Academy of Sciences was established in November 1949.

The first National Conference of Chinese Mathematic Institutes was held in August 1951, discussing the developmental trend and problems in the math education reform of various schools.

Math research progressed significantly after 1949. At the beginning of the 1950s, a lot of monographs by noted mathematicians likeHua Luogeng,Su Buqing, Chen Jiangong, and Li Yan, were published. By 1966, about 20,000 mathematic dissertations had already been released, and many of these works being at the world's leading level. At the same time, many outstanding mathematicians like the abovementioned four were brought up during this time.

In the latter half of the 1960s (the first part of the 1966-1976 Culture Revolution), the research of Chinese math was suspended, with the education paralyzed, talents lost, and external communications halted. The condition improved slightly later with the efforts of many.

In 1973, Chen Jingrui published a dissertation in the China Science magazine, making a major breakthrough of the Goldbach's conjecture (one of the oldest unsolved number theory problems) by using the methods of sieve theory (a way to estimate the size of sifted integer sets).

In November 1978, the Third National Conference of Chinese Mathematic Institutes was held, symbolizing the resurgence of Chinese math.

In 1986, famous mathematicianWu Wenjuntook part in the International Congress of Mathematicians for the first time and delivered a 45-minute speech on the history of ancient Chinese math.

In the 50th anniversary of the establishment of the Chinese Mathematic Institute in 1985, a long-term development objective was set for Chinese math. As a result, in the past couple of years, the number of dissertations has doubled and the quality also improved significantly.

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