Yang Hui
Encyclopedia
Yang Hui courtesy name Qianguang (谦光), was a Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
from Qiantang (modern Hangzhou
Hangzhou
Hangzhou , formerly transliterated as Hangchow, is the capital and largest city of Zhejiang Province in Eastern China. Governed as a sub-provincial city, and as of 2010, its entire administrative division or prefecture had a registered population of 8.7 million people...
), Zhejiang
Zhejiang
Zhejiang is an eastern coastal province of the People's Republic of China. The word Zhejiang was the old name of the Qiantang River, which passes through Hangzhou, the provincial capital...
province during the late Song Dynasty
Song Dynasty
The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...
(960-1279 AD). Yang worked on magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
s, magic circles
Magic circle (mathematics)
Magic circles were invented by the Song Dynasty Chinese mathematician Yang Hui . It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical...
and the binomial theorem
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
, and is best known for his contribution of presenting 'Yang Hui's Triangle'. This triangle was the same as Pascal's Triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
, discovered by Yang's predecessor Jia Xian
Jia Xian
Jia Xian was a Chinese mathematician from Kaifeng of the Song Dynasty, in the first half of the 11th century.-Biography:...
(贾宪). Yang was also a contemporary to the other famous mathematician Qin Jiushao.
Written work
The earliest extant Chinese illustration of 'Pascal's TrianglePascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
' is from Yang's book Xiangjie Jiuzhang Suanfa (详解九章算法) 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 Jia Xian
Jia Xian
Jia Xian was a Chinese mathematician from Kaifeng of the Song Dynasty, in the first half of the 11th century.-Biography:...
who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as Ruji Shisuo (如积释锁) or Piling-up Powers and Unlocking Coefficients, which is known through his contemporary mathematician Liu Ruxie (刘汝谐). 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
Zhu Shijie , courtesy name Hanqing , pseudonym Songting , was one of the greatest Chinese mathematicians lived during the Yuan Dynasty....
's book Jade Mirror of the Four Unknowns (四元玉鉴) of 1303 AD.
Around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa (续古摘奇算法) and the Suanfa Tongbian Benmo (算法通变本末)(summarily called Yang Hui suan fa 杨辉算法). In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as magic circles
Magic circle (mathematics)
Magic circles were invented by the Song Dynasty Chinese mathematician Yang Hui . It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical...
and vertical-horizontal diagram
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...
s of complex combinatorial arrangements known as magic squares' and magic circles
Magic circle (mathematics)
Magic circles were invented by the Song Dynasty Chinese mathematician Yang Hui . It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical...
, providing rules for their construction. In his writing, he harshly criticized the earlier works of Li Chunfeng
Li Chunfeng
Li Chunfeng was a Chinese mathematician, astronomer, and historian who was born in today's Baoji, Shaanxi during the Sui and Tang dynasties. He was first appointed to the Imperial Astronomy Bureau to help institute a calendar reform. He eventually ascended to deputy of the Imperial Astronomy...
(李淳風) and Liu Yi (刘益), 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
s 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
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon
Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
. There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci
Matteo Ricci
Matteo Ricci, SJ was an Italian Jesuit priest, and one of the founding figures of the Jesuit China Mission, as it existed in the 17th-18th centuries. His current title is Servant of God....
and the Ming
Ming Dynasty
The Ming Dynasty, also Empire of the Great Ming, was the ruling dynasty of China from 1368 to 1644, following the collapse of the Mongol-led Yuan Dynasty. The Ming, "one of the greatest eras of orderly government and social stability in human history", was the last dynasty in China ruled by ethnic...
official Xu Guangqi
Xu Guangqi
Xu Guangqi , was a Chinese scholar-bureaucrat, agricultural scientist, astronomer, and mathematician in the Ming Dynasty. Xu was a colleague and collaborator of the Italian Jesuits Matteo Ricci and Sabatino de Ursis and they translated several classic Western texts into Chinese, including part of...
in the early 17th century.
Yang's writing represents the first in which quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. 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.