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Trigonometry The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.[29] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[29] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc.[42] Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[43] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[29][44] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:
Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).[45] Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).

Movie Title Year Distributor Notes Rev Formats Star Virgin 1979 VCX Bald 1 DRO Best of VCX Classics 2 2005 VCX There are many summation series equations given without proof in the Mirror. A few of the summation series are:[40] {\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}}1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n + 1)(2n + 1)\over 3!} {\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}}1 + 8 + 30 + 80 + \cdots + {n^2(n + 1)(n + 2)\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\over 5!} Mathematical Treatise in Nine Sections Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (ca. 1202 – ca. 1261 AD) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.[38] Magic squares and magic circles The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten.[41] He also worked with magic circle. Ming mathematics After the overthrow of the Yuan Dynasty, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes: At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century.[47] Correspondingly, scholars paid less attention to mathematics; pre-eminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the Tian yuan shu (Increase multiply) method.[48] Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To the average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art, he omitted Tian yuan shu and the increase multiply method.[49][failed verification] An abacus. Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its suan pan form. Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. Zhusuan, the arithmetic calculation through abacus, inspired multiple new works. Suanfa Tongzong (General Source of Computational Methods), a 17-volume work published in 1592 by Cheng Dawei, remained in use for over 300 years.[50] Zhu Zaiyu, Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy, a precision that enabled his development of the equal-temperament system. Although this switch from counting rods to the abacus allowed for reduced computation times, it may have also led to the stagnation and decline of Chinese mathematics. The pattern rich layout of counting rod numerals on counting boards inspired many Chinese inventions in mathematics, such as the cross multiplication principle of fractions and methods for solving linear equations. Similarly, Japanese mathematicians were influenced by the counting rod numeral layout in their definition of the concept of a matrix. Algorithms for the abacus did not lead to similar conceptual advances. (This distinction, of course, is a modern one: until the 20th century, Chinese mathematics was exclusively a computational science.[51]) In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi, he was able to translate Euclid's Elements using the same techniques used to teach classical Buddhist texts.[52] Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition.[53] However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone.[54] Qing dynasty Under the Western-educated Kangxi Emperor, Chinese mathematics enjoyed a brief period of official support.[55] At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume Shuli Jingyun [The Essence of Mathematical Study] (printed 1723) which gave a systematic introduction to western mathematical knowledge.[56] At the same time, Mei Goucheng also developed to Meishi Congshu Jiyang [The Compiled works of Mei]. Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending (1633-1721), Goucheng's grandfather.[57][58] The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations.[59] However, no sooner were the encyclopedias published than the Yongzheng Emperor acceded to the throne. Yongzheng introduced a sharply anti-Western turn to Chinese policy, and banished most missionaries from the Court. With access to neither Western texts nor intelligible Chinese ones, Chinese mathematics stagnated. In 1773, the Qianlong Emperor decided to compile Siku Quanshu (The Complete Library of the Four Treasuries). Dai Zhen (1724-1777) selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties.[60] The long-missing mathematical works from Song and Yuan dynasties such as Si-yüan yü-jian and Ceyuan haijing were also found and printed, which directly led to a wave of new research.[61] The most annotated work were Jiuzhang suanshu xicaotushuo (The Illustrations of Calculation Process for The Nine Chapters on the Mathematical Art ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin.[62] Western influences In 1840, the First Opium War forced China to open its door and looked at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements and 13 volumes on Algebra.[63][64] With the assistance of Joseph Edkins, more works on astronomy and calculus soon followed. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? But by the end of the century, it became clear that China could only begin to recover its sovereignty by incorporating Western works. Chinese scholars, taught in Western missionary schools, from (translated) Western texts, rapidly lost touch with the indigenous tradition. As Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China."[65] Western mathematics in modern China Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields. Some famous modern ethnic Chinese mathematicians include: Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century and was awarded the Wolf prize for his immense number of mathematical contributions.[66][67] Ky Fan, made a tremendous number of fundamental contributions to many different fields of mathematics. His work in fixed point theory, in addition to influencing nonlinear functional analysis, has found wide application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations. Shing-Tung Yau, his contributions have influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics and subsequently awarded the Fields medal for his contributions. Terence Tao, an ethnic Chinese child prodigy who received his master's degree at age 16, was the youngest participant in the International Mathematical Olympiad's entire history, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history. He went on to receive the Fields medal. Yitang Zhang, a number theorist who established the first finite bound on gaps between prime numbers. Chen Jingrun, a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes) which is now called Chen's theorem .[68] His work was known as a milestone in the research of Goldbach's conjecture. Mathematics in the People's Republic of China In 1949, at the beginning of the founding of the People's Republic of China, the government paid great attention to the cause of science although the country was in a predicament of lack of funds. The Chinese Academy of Sciences was established in November 1949. The Institute of Mathematics was formally established in July 1952. Then, the Chinese Mathematical Society and its founding journals restored and added other special journals. In the 18 years after 1949, the number of published papers accounted for more than three times the total number of articles before 1949. Many of them not only filled the gaps in China's past, but also reached the world's advanced level.[69] During the chaos of the Cultural Revolution, the sciences declined. In the field of mathematics, in addition to Chen Jingrun, Hua Luogeng, Zhang Guanghou and other mathematicians struggling to continue their work. After the catastrophe, with the publication of Guo Moruo's literary "Spring of Science", Chinese sciences and mathematics experienced a revival. In 1977, a new mathematical development plan was formulated in Beijing, the work of the mathematics society was resumed, the journal was re-published, the academic journal was published, the mathematics education was strengthened, and basic theoretical research was strengthened.[69] An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988. When there are some initial states of N celestial bodies, one of the celestial bodies ran to infinity or speed in a limited time. Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013.[70] In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree. This conjecture can be traced back to Fatou in the 1920s, and later Smale proposed him in the 1960s. Axiom A, and guess that the hyperbolic system should be dense in any system, but this is not true when the dimension is greater than or equal to 2, because there is homoclinic tangencies. The work of Shen Weixiao and others is equivalent to confirming that Smale's conjecture is correct in one dimension. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade.[70] Performance at the IMO In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and the won the all-members-gold IMO with a full team the most number of times

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