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Liu Hui  

Traditional Chinese  劉徽  

Liu Hui (fl. 3rd century CE) was a Chinese mathematician and writer who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art , in which he was possibly the first mathematician to discover, understand and use negative numbers. He was a descendant of the Marquis of Zi District (菑鄉侯) of the Eastern Han dynasty, whose marquisate is in presentday Zichuan District, Zibo, Shandong. He completed his commentary to the Nine Chapters in the year 263. He probably visited Luoyang, where he measured the sun's shadow.
Along with Zu Chongzhi (429–500), Liu Hui was known as one of the greatest mathematicians of ancient China.^{ [2] } Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 12381298 AD) expressed his mathematical results in full decimal expressions.^{ [3] }^{ [4] }
Liu provided commentary on a mathematical proof of a theorem identical to the Pythagorean theorem.^{ [5] } Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".^{ [6] }
In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.^{ [7] } He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented:
Liu Hui also presented, in a separate appendix of 263 AD called Haidao Suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers.^{ [12] } This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them".^{ [13] } With this, the following cases are considered in his work:
Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.^{ [14] } Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.^{ [15] }
Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.
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Yang Hui, courtesy name Qianguang (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang, Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.
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Haidao Suanjing was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of The Nine Chapters on the Mathematical Art. During the Tang Dynasty, this appendix was taken out from The Nine Chapters on the Mathematical Art as a separate book, titled Haidao suanjing (Sea Island Mathematical Manual), named after problem No 1 "Looking at a sea island." In the time of the early Tang dynasty, Haidao Suanjing was selected into one of The Ten Computational Canons as the official mathematical texts for imperial examinations in mathematics.
Liu Hui's π algorithm was invented by Liu Hui, a mathematician of the Cao Wei Kingdom. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits. Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96gon provided an accuracy of five digits: π ≈ 3.1416.
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