Zhu Shijie

Zhu Shijie (Chinese: 朱世杰; pinyin: Zhū Shìjié; Wade–Giles: Chu Shih-chieh, 1249–1314), courtesy name Hanqing (汉卿), pseudonym Songting (松庭), was one of the greatest Chinese mathematicians living during the Yuan Dynasty.
Zhu was born close to today's Beijing. Two of his mathematical works have survived. Introduction to Computational Studies (算学启蒙Suanxue qimeng), and Jade Mirror of the Four Unknowns.

Suanxue qimeng

Suanxue Qimeng (算学启蒙), written in 1299, is an elementary textbook on mathematics in three volumes, 20 chapters and 259 problems. This book also showed how to measure different two-dimensional shapes and three-dimensional solids. The Introduction had an important influence on the development of mathematics in Japan. The book was once lost in China until Qing dynasty mathematician Luo Shilin bought a Korean printed edition, and re-published in Yangzhou. Since then this book was reprinted several times.

Illustrations in Jade Mirror of the Four Unknowns

Zhu's second book, Jade Mirror of the Four Unknowns, written in 1303, is his most important work. With this book, Zhu brought Chinese algebra to its highest level. The first four of the 288 problems for solution 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 solved the high order equation by Southern Song dynasty mathematician Qin Jiushao's "Ling long kai fang" method published in Shùshū Jiǔzhāng (“Mathematical Treatise in Nine Sections”) in 1247 (more than 570 years before English mathematician William Horner's method using synthetic division). To do this, he makes use of the Pascal triangle, which he labels as the diagram of an ancient method first discovered by Jia Xian before 1050. The final equation and one of its solutions is 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 be reducing the matrix of their coefficients to diagonal form. His methods pre-date Blaise Pascal, William Horner, and modern matrix methods by many centuries. The preface of the book describes how Zhu travelled around China for 20 years as a teacher of mathematics.

The methods of Jade Mirror of the Four Unknowns form the foundation for Wu's method of characteristic set.