Chinese mathematics in the context of "Diophantine approximation"

Aside from many original inventions, the Chinese were also early original pioneers in the discovery of natural phenomena which can be found in the human body, the environment of the world, and the immediate Solar System. They also discovered many concepts in mathematics. The list below contains discoveries which found their origins in China.

Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.

Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega).

Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Song dynasty (960–1279). Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).

In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination).

The Dream Pool Essays (or Dream Torrent Essays) was an extensive book written by the Chinese polymath and statesman Shen Kuo (1031–1095), published in 1088 during the Song dynasty (960–1279) of China. Shen compiled this encyclopedic work while living in forced retirement from government office, naming the book after his private estate near modern Zhenjiang, Jiangsu province. The Dream Pool Essays was heavily reorganized in reprint editions by later Chinese authors from the late 11th to 17th centuries. In modern times it has been translated from Chinese into several languages. These include English, German, French, and Japanese translations.

The Dream Pool Essays covers a range of topics including discoveries and advancements in Traditional Chinese medicine, mathematics, astronomy, science and technology, optics, architecture and civil engineering, metallurgy, and early archaeology. Observations of the natural world included those of wildlife, meteorology, hypotheses advancing early ideas in geomorphology and climate change based on findings of petrification and natural erosion, and strange recorded phenomena such as the description of an unidentified flying object. In addition to establishing the theory of true north in magnetic declination towards the North Pole, Shen was also the first to record the use of a compass for navigation, the first to describe the invention of movable type printing by contemporary artisan Bi Sheng, and the first in China to describe a drydock for repairing boats out of water.

The Book on Numbers and Computation (Chinese: 筭數書; pinyin: Suàn shù shū), or the Writings on Reckoning, is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC. It was preserved among the Zhangjiashan Han bamboo texts and contains similar mathematical problems and principles found in the later Eastern Han period text of The Nine Chapters on the Mathematical Art.

The Song dynasty (Chinese: 宋朝; 960–1279 CE) witnessed many substantial scientific and technological advances in Chinese history. Some of these advances and innovations were the products of talented statesmen and scholar-officials drafted by the government through imperial examinations. Shen Kuo (1031–1095), author of the Dream Pool Essays, is a prime example, an inventor and pioneering figure who introduced many new advances in Chinese astronomy and mathematics, establishing the concept of true north in the first known experiments with the magnetic compass. However, commoner craftsmen such as Bi Sheng (972–1051), the inventor of movable type printing (in a form predating the printing press of Johannes Gutenberg), were also heavily involved in technical innovations.

The ingenuity of advanced mechanical engineering had a long tradition in China. The Song engineer Su Song, who constructed a hydraulically-powered astronomical clocktower, admitted that he and his contemporaries were building upon the achievements of the ancients such as Zhang Heng (78–139), an astronomer, inventor, and early master of mechanical gears whose armillary sphere was automatically rotated by a waterwheel and clepsydra timer. The application of movable type printing advanced the already widespread use of woodblock printing to educate and amuse Confucian students and the masses. The application of new weapons employing the use of gunpowder enabled the Song to ward off its militant enemies—the Liao, Western Xia, and Jin with weapons such as cannons—until its collapse to the Mongol forces of Kublai Khan in the late 13th century.