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Zhu Shijie

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Zhu Shijie was a major Chinese mathematician during the Yuan Dynasty.

朱世杰算学启蒙中的天元术
朱世杰算学启蒙中的天元术

He lived from 1249 to 1314. He was born near the area of modern-day Beijing. Zhu was not just a thinker; he was also a dedicated educator. The preface of his most famous book says he traveled across China for 20 years to teach mathematics. His work helped move the field of algebra forward by many centuries. He created systems to solve complex problems that were much more advanced than many other methods of his time.

Zhu left behind two major surviving works. The first was the Introduction to Computational Studies, written in 1299.

朱世杰算学启蒙中的天元术
朱世杰算学启蒙中的天元术

This was an elementary textbook divided into three volumes. It contained 20 chapters and 259 math problems. This book taught students how to measure two-dimensional shapes and three-dimensional solids. It was so effective that it later influenced the development of mathematics in Japan. In China, the book was once lost until a mathematician named Luo Shilin found a Korean edition and republished it.

His second and most important work was the Jade Mirror of the Four Unknowns, written in 1303.

SIYUAN YUJIAN PDF-036-036
SIYUAN YUJIAN PDF-036-036

This book focused on advancing Chinese algebra. It contains 288 solved problems. Each problem includes the final equation and one of its solutions. This work introduced a sophisticated method for handling multiple unknown values.

四元自乘演段图
四元自乘演段图

He used up to four unknowns to solve these problems. He gave these unknowns specific names: Heaven, Earth, Man, and Matter.

The mechanism of his algebraic method was very logical. First, he would convert a problem stated in words into a system of polynomial equations. These equations could reach up to the 14th order. Next, he used a process of successive elimination. This allowed him to reduce the complex system down to a single polynomial equation with only one unknown. To solve these high-order equations, he used the ling long kai fang method. This method was developed by the Southern Song mathematician Qin Jiushao in 1247. This process allowed him to find solutions that were incredibly difficult to reach.

Zhu Shijie utilized many mathematical patterns that are well-known today. He made use of what is now called Pascal's triangle. He noted that the mathematician Jia Xian had already discovered this pattern before the year 1050. Zhu used these patterns to understand series and progressions. He classified these series based on the coefficients found in the triangle. He also worked with square and cube roots. He found these roots by solving quadratic and cubic equations.

His mathematical insights were far ahead of his global contemporaries. For example, his techniques were used more than 570 years before the English mathematician William Horner used synthetic division. He also developed ways to solve systems of linear equations. He did this by reducing the matrix of their coefficients to a diagonal form. This was a very early version of modern matrix methods. His work with algebraic equations also involved using a version of the resultant.

The legacy of Zhu Shijie connects deeply to modern mathematical systems. His advanced methods in the Jade Mirror of the Four Unknowns helped form the foundation for the Wu method of characteristic set. By solving complex polynomial equations and understanding the relationship between coefficients and patterns, he helped build the tools used in algebra today. His ability to turn verbal descriptions into precise mathematical structures remains a fundamental part of how we study math.

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