Zhu Shijie (, 1249–1314),

courtesy name A courtesy name ( zh, s=字, p=zì, l=character), also known as a style name, is an additional name bestowed upon individuals at adulthood, complementing their given name. This tradition is prevalent in the East Asian cultural sphere, particula ...

Hanqing (),

pseudonym A pseudonym (; ) or alias () is a fictitious name that a person assumes for a particular purpose, which differs from their original or true meaning ( orthonym). This also differs from a new name that entirely or legally replaces an individual's o ...

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

Yuan Dynasty The Yuan dynasty ( ; zh, c=元朝, p=Yuáncháo), officially the Great Yuan (; Mongolian language, Mongolian: , , literally 'Great Yuan State'), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after Div ...

. Zhu was born close to today's Beijing. Two of his mathematical works have survived: ''Introduction to Computational Studies'' ( ''Suan-hsüeh Ch'i-mong'') and '' Jade Mirror of the Four Unknowns'' ( ''Ssŭ-yüan Yü-chien'').

''Suanxue Qimeng''

The ''Suanxue Qimeng'' (), written in 1299, is an elementary textbook on mathematics in three volumes, 20 chapters and 259 problems. This book also showed how to measure two-dimensional shapes and three-dimensional solids. The ''Introduction'' strongly influenced the development of mathematics in Japan. The book was once lost in China until the Qing dynasty mathematician Luo Shilin bought a Korean printed edition and republished it in Yangzhou.

''Jade Mirror of the Four Unknowns''

Zhu's second book, '' Jade Mirror of the Four Unknowns'' (1303) was his most important work, advancing Chinese algebra. The first four of the 288 solved problems illustrate his method of the four unknowns. He shows how to convert a problem stated verbally into a system of polynomial equations (up to 14th order), by using up to four unknowns: 天 Heaven, 地 Earth, 人 Man, 物 Matter, and then how to reduce the system to a single polynomial equation in one unknown by successive elimination of unknowns. He then solves the high order equation by the ''ling long kai fang'' (玲瓏開方) method of

Southern Song The Song dynasty ( ) was an imperial dynasty of China that ruled from 960 to 1279. The dynasty was founded by Emperor Taizu of Song, who usurped the throne of the Later Zhou dynasty and went on to conquer the rest of the Ten Kingdoms, ending ...

dynasty 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 ...

(from ''Shùshū Jiǔzhāng,'' “

Mathematical Treatise in Nine Sections The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of th ...

” of 1247). This was more than 570 years before English mathematician William Horner's method using synthetic division. Zhu makes use of what is currently known as

Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

, which he refers to as discovered by Jia Xian before 1050. The final equation and one of its solutions are given for each of the 288 problems. Zhu also found square and cube roots by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle. He also showed how to solve systems of

linear equations In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...

by reducing the matrix of their coefficients to

diagonal form In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is of the form :\sum_^n a_i ^m\ for some degree ''m''. Such forms ''F'', and the hypersurfaces ...

. He moreover applied these methods to

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equati ...

s, using a version of the

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...

.Mumford 2010, p. 122. His methods pre-date

Blaise Pascal Blaise Pascal (19June 162319August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest ...

, William Horner, and modern matrix methods by many centuries. The preface of the book describes how Zhu traveled China for 20 years teaching mathematics. The methods of ''Jade Mirror of the Four Unknowns'' form the foundation for Wu's method of characteristic set.

References

*Du, Shiran
"Zhu Shijie"
''

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, be ...

'' (Mathematics Edition), 1st ed. *GRATTAN-GUINNESS, I.: ''The Norton History of the Mathematical Sciences'', 1998. *Guo Shuchun (tr. modern Chinese), Chen Zaixin (English tr.), Guo Jinhai (annotation), Zhu Shijie: ''Jade mirror of the Four Unknowns'', Chinese and English bilingual, vol I & 2, Liaoning education Press, China, 2006. *HO Peng-Yoke: Article on Chu Shih-chieh in the ''Dictionary of Scientific Biography'', New York, 1970.hi *Hoe, J.: ''The jade mirror of the four unknowns'', Mingming Bookroom, New Zealand, 2007. *Hoe, J.: ''Les systèmes d'équations polynômes dans le Siyuan Yujian (1303)'', Paris, Collège de France (Mémoires de l'Institut des Hautes Etudes Chinoises, Vol VI),1977. *KONANTZ, E.L.:The Precious Mirror of the Four Elements, ''China journal of Science and Arts'', Vol 2, No 4, 1924. *LAM Lay-yong: Chu shih-chieh's Suan hsüeh ch'i-meng, ''Archive for the history of sciences'', Vol 21, Berlin, 1970. *MARTZLOFF, J-C.: ''A history of Chinese Mathematics'', Springer-Verlag, Berlin, 1997. * MIKAMI Yoshio, Development of Mathematics in China and Japan, Chapter 14 Chu Shih-chieh p89-98. 1913 Leipzig. Library of Congress catalog card number 61-13497. *Mumford, David, "What’s so Baffling About Negative Numbers? — a Cross-Cultural Comparison", in C. S. Seshadri (Ed.), ''Studies in the History of Indian Mathematics'', 2010.

External links

* {{DEFAULTSORT:Zhu, Shijie 1249 births 1314 deaths 13th-century Chinese mathematicians 14th-century Chinese mathematicians Educators from Hebei Mathematicians from Imperial China Yuan dynasty science writers Writers from Hebei