Yang Hui (, ca. 1238–1298),

courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere, including China, Japan, Korea, and Vietnam.Ulrich Theob ...

Qianguang (), was a Chinese mathematician and writer during the

Song dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...

. Originally, from Qiantang (modern

Hangzhou Hangzhou ( or , ; , , Standard Mandarin pronunciation: ), also romanized as Hangchow, is the capital and most populous city of Zhejiang, China. It is located in the northwestern part of the province, sitting at the head of Hangzhou Bay, whic ...

Zhejiang Zhejiang ( or , ; , Chinese postal romanization, also romanized as Chekiang) is an East China, eastern, coastal Provinces of China, province of the People's Republic of China. Its capital and largest city is Hangzhou, and other notable citie ...

), Yang worked on magic squares, magic circles and the

binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as

Pascal's Triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...

, discovered by Yang's predecessor

Jia Xian Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty. Biography According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...

. Yang was also a contemporary to the other famous mathematician

Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gau ...

Written work

The earliest extant Chinese illustration of '

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...

' is from Yang's book ''Xiangjie Jiuzhang Suanfa'' ()Fragments of this book was retained in the Yongle Encyclopedia vol 16344, in British Museum Library 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 mathematician

Needham, Volume 3, 134-137. who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as ''Rújī Shìsuǒ'' () or ''Piling-up Powers and Unlocking Coefficients'', which is known through his contemporary mathematician Liu Ruxie ().Needham, Volume 3, 137. Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients). It appeared again in a publication of

Zhu Shijie Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...

's book ''Jade Mirror of the Four Unknowns'' () of 1303 AD.Needham, Volume 3, 134-135. Around 1275 AD, Yang finally had two published mathematical books, which were known as the ''Xugu Zhaiqi Suanfa'' () and the ''Suanfa Tongbian Benmo'' (, summarily called Yang Hui suanfa ).Needham, Volume 3, 104. In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as magic circles and vertical-horizontal diagrams of complex

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

arrangements known as magic squares, providing rules for their construction.Needham, Volume 3, 59-60. In his writing, he harshly criticized the earlier works of Li Chunfeng and Liu Yi (), the latter of whom were both content with using methods without working out their theoretical origins or principle. Displaying a somewhat modern attitude and approach to mathematics, Yang once said: :''The men of old changed the name of their methods from problem to problem, so that as no specific explanation was given, there is no way of telling their theoretical origin or basis.'' In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another. This was the same idea expressed in the Greek mathematician

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon. There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system.Needham, Volume 3, 105. However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci and the

Ming The Ming dynasty (), officially the Great Ming, was an imperial dynasty of China, ruling from 1368 to 1644 following the collapse of the Mongol-led Yuan dynasty. The Ming dynasty was the last orthodox dynasty of China ruled by the Han peop ...

official

Xu Guangqi Xu Guangqi or Hsü Kuang-ch'i (April 24, 1562– November 8, 1633), also known by his baptismal name Paul, was a Chinese agronomist, astronomer, mathematician, politician, and writer during the Ming dynasty. Xu was a colleague and collaborato ...

in the early 17th century.Needham, Volume 3, 106. Yang's writing represents the first in which

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...

s with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.Needham, Volume 3, 46. 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 3 ⁴⁄₁₀ ft. and length of 36 paces 2 ⁸⁄₁₀, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.Needham, Volume 3, 45.

Notes

References

*Needham, Joseph (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd. *Li, Jimin
"Yang Hui"
''

Encyclopedia of China The ''Encyclopedia of China'' () is the first large-entry modern encyclopedia in the Chinese language. The compilation began in 1978. Published by the Encyclopedia of China Publishing House, the encyclopedia was issued one volume at a time, begin ...

'' (Mathematics Edition), 1st ed.

External links

Yang Hui at MacTutor
{{DEFAULTSORT:Yang, Hui 1238 births 1298 deaths 13th-century Chinese mathematicians Magic squares Mathematicians from Zhejiang Medieval Chinese mathematicians Song dynasty science writers Writers from Hangzhou

Written work

See also

Notes

References

External links