Yang Hui (simplified Chinese:杨辉;traditional Chinese:楊輝;pinyin:Yáng Huī, ca. 1238–1298),courtesy nameQianguang (謙光), was a Chinese mathematician and writer during theSong dynasty. Originally, from Qiantang (modernHangzhou,Zhejiang), Yang worked onmagic squares,magic circles and thebinomial theorem, and is best known for his contribution of presentingYang Hui's Triangle. This triangle was the same asPascal's Triangle, discovered by Yang's predecessorJia Xian. Yang was also a contemporary ofQin Jiushao, another well-known Chinese mathematician.
The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's bookXiángjiě Jiǔzhāng Suànfǎ (詳解九章算法)[1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematicianJia Xian[2] who expounded it around 1100 AD, about 500 years before Pascal. His book (now lost), known asRújī Shìsuǒ (如積釋鎖) orPiling-up Powers and Unlocking Coefficients, was known through his contemporary mathematicianLiu Ruxie (劉汝諧).[3] Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients).[3] It appeared again in a publication ofZhu Shijie's bookJade Mirror of the Four Unknowns (四元玉鑒) of 1303 AD.[4]
Around 1275 AD, Yang finally had two published mathematical books, which were known as theXùgǔ Zhāijī Suànfǎ (續古摘奇算法) and theSuànfǎ Tōngbiàn Běnmò (算法通變本末, summarily calledYáng Huī Suànfǎ楊輝算法).[5] In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known asmagic circles and vertical-horizontaldiagrams of complexcombinatorial arrangements known asmagic squares, providing rules for their construction.[6] In his writing, he harshly criticized the earlier works ofLi Chunfeng andLiu Yi (劉益), the latter of whom were both content with using methods without working out their theoretical origins or principle.[5] Displaying a somewhat modern attitude and approach tomathematics, Yang once said:
In his written work, Yang provided theoretical proof for the proposition that the complements of theparallelograms which are about the diameter of any given parallelogram are equal to one another.[5] This was the same idea expressed in the Greek mathematicianEuclid's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle andgnomon.[5] There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system.[7] However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian JesuitMatteo Ricci and theMing officialXu Guangqi in the early 17th century.[8]
Yang's writing represents the first in whichquadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[9] Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 34⁄10 ft. and length of 36 paces 28⁄10, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.[10]
The Yang-Hui Award is presented to mathematicians or scientists who have gained international recognition for their exceptional contributions throughout their careers.[11] It was awarded to Salvatore Capozziello for his work withNoether symmetries;Mahouton Norbert Hounkonnou for his work in deformed quantum algebras; and to Delfim F. M. Torres for his mathematical modelling ofCOVID-19 in 2023 at the International Conference on Mathematical Analysis, Applications and Computational Simulation (ICMAACS 2023), Shanghai, China, November 22-26, 2023.[12][13]
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