At the end of the 13th century, China was unified by the Yuan Dynasty, and the damaged economy and culture quickly prospered again. In order to rejuvenate the country, Mongolian rulers began to respect knowledge and select a large number of talents, pushing the development of various sciences to a new height.

At that time, Kublai Khan gathered a large number of Han intellectuals to form think tanks, among them Wang Ji, Guo Shoujing, Li Zhi and others. The characters in this think tank were proficient in mathematics and calendar.

At this time, Zhu Shijie also inherited the main achievement of northern mathematics-Tianyuanshu, and extended it from binary and ternary to the solution of quaternion equations. In addition to accepting the mathematical achievements of the North, Zhu Shijie also absorbed the mathematical achievements of the South, especially various daily algorithms, commercial arithmetic, and popular songs.

Before the Yuan and the Southern Song dynasties, the communication between the North and the South, especially the academic communication, was almost cut off. Mathematicians in the South knew nothing about the North s astronomy, and Mathematicians in the North were rarely influenced by the South. Zhu Shijie once traveled around the Quartet. After more than 20 years of study tours and lectures, he finally engraved his two masterpieces of mathematics— Enlightenment of Mathematical Studies and Yuanjian of Four Elements in Yangzhou in 1299 and 1303 .

Enlightenment in Mathematics includes all aspects of mathematics at that time from multiplication and division operations and shortcut algorithms to square roots, celestial elements, equations, etc., from shallow to deep, forming a more complete system. In front of the text, there are 18 items including Jiu Jiu Jiu Ji, Jiu Ge Jiu Ju, Jin Liang Hua Zhe Ge, the calculation system, the big and small round method, the weights and measures system, the circumference ratio, the law of plus and minus multiplication, and the rule of opening. As a summary, as the preliminaries of the book, the law of multiplication of positive and negative numbers not only appeared in Chinese mathematical works, but also appeared in the world for the first time. Many of the song formulas are more complete and accurate than Yang Hui s, and some are almost completely consistent with modern abacus formulas.

Siyuan Yujian is Zhu Shijie s most outstanding work. In this book, he recorded his opinions on the solutions of multivariate higher-order equations, the sum of higher-order difference series, and the method of higher-order interpolation. Researchers of the history of mathematics highly appraise it as one of the most important and most important mathematical works in ancient Chinese mathematical science works.

Zhu Shijie s other major contribution is the research on stacking . He researched a series of new series-shaped series summation problems, and summarized the formula of triangular stacks from it. In fact, he obtained this kind of systematic and universal summation problems of arbitrary high-order series of equal-level differences. solution. Zhu Shijie also referred to the triangular stacking formula to the judgment method, and pointed out that the coefficients in the formula are exactly the product of the triangular stacks in this order, so that a four-fold difference-containing formula is obtained. He also extended this formula to a formula containing arbitrarily high differences, which is the first time in the history of world mathematics.

In the history of Chinese mathematics, Zhu Shijie formally proposed the correct rule of positive and negative multiplication for the first time; he discussed the calculation of the surface area of spheres. This is the only discussion in ancient Chinese mathematics books. Although the conclusion is incorrect, it is innovative. Spirit is precious. In Enlightenment of Arithmetic, he recorded the complete nine return division formula, which is almost identical to the abacus return formula that is now circulating.