Contributions of Zhu Shijie

The main contribution of Zhu Shijie was the creation of a complete method of eliminating unknowns, called the quadratic elimination method. This method was the world leader for a long time until the 18th century, when the French mathematician Bezout proposed a general method for solving systems of higher equations, which surpassed Zhu Shijie. In addition to the quadratic technique, there are two other important achievements in the Quadratic Yujian, i.e., the creation of the general higher-order equidistant progression summation formula and the equidistant quadratic interpolation formula, the latter of which is commonly known as the beckoning technique. This book represents the highest level of Song and Yuan mathematics, and the American historian of science G. Sarton praised it as "one of the most important of the Chinese mathematical works, and one of the outstanding mathematical works of the Middle Ages". Zhu Shijie was at the height of the development of traditional Chinese mathematics, when society "honored arithmetic, and the subject was growing" and mathematical works were widely disseminated.

Solutions of multivariate systems of higher-order equivariant equations, higher-order sums, and higher-order interpolations were all studied in depth, and he authored Arithmetic Enlightenment (1299) and Quaternary Yujian (1303), each in three volumes, in the latter of which solutions of higher-order systems of associative equations of up to four elements were discussed, and the expression and operation of polynomials linked together with elimination methods, which were already approaching the modern algebra, were at the leading position in the world. He was in the leading position in the world, and his knowledge of the formula of the higher-order invocation method was four hundred years earlier than that of the West, and Chinese and foreign historians of mathematics have highly praised Zhu Shijie and his famous book, "Quaternion Yujian".

The extension of the technique from Tianyuan to the system of higher-order conjugate equations of binary, ternary and quaternary is another outstanding creation of the Song-Yuan mathematicians. The one that has survived to this day and provides a systematic discussion of this outstanding creation is Zhu Shijie's Four Elements of Yujian. The book was written in 1303. It consists of 3 volumes, 24 doors, and 288 questions, and mainly deals with the solution of higher order systems of equations (which is also Zhu Shijie's greatest contribution), the summation of higher order equidistant series, and the method of higher order interpolation. It is an important representative work that has been handed down to the present day and provides a systematic treatment of quaternions.

Based on Tianyuan, Zhu Shijie established the "theory of quadratic higher order equations", in which he placed the constant term in the center (i.e., "too"), and then "set up the heavenly element at the bottom, the earthly element at the left, the human element at the right, and the material element at the top". and things on one in the upper", "heaven, earth, people, things," the four "yuan" on behalf of the unknowns, (i.e., the equivalent of today's x, y, z, w,) the powers of the four elements in the upper, lower, left, right four directions, the other items were placed in the four quadrants. If we use the modern x, y, z, w to represent the sky, earth, people and things, then we can express the method of Zhu Shijie's high multiple equations: the two graphs above, "quadratic quadratic chips" and "quadratic quadratic chips" represent the equations are: x + y + z + w = 0, and the quadratic quadratic chips are: x + y + z + w = 0, and the quadratic quadratic chips are: x + y + z + w = 0. y+z+w=0,

Using the above method to list the quaternary equations, and then the joint system of equations to solve the system of equations, the method is to use the elimination method to answer, first choose a dollar for the unknown, the other elements of the composition of the polynomial as the coefficients of the unknown, and then the quaternary quadratic elimination of a dollar, into the triple triple, and then eliminated a dollar into the binary two-way, and then eliminated a dollar, you get the day only contains a dollar of the dollar to open! way, and then use the increasing multiplication open method to find the positive root. This is a major development in the linear method of solving systems of equations. In the West, a more systematic study of multivariate systems of equations had to wait until the 16th century. Summation of higher-order equivariant series and higher-order interpolation are also important contents of the Quadratic Yujian. Formulas can be generalized from a series of trigonometric pallet formulas in many summation problems. Zhu Shijie gives the formula in the above equation when p = 1, 2, ......6. In addition, there are other higher order equidistant series summation formulas. In terms of the trick difference method, Zhu Shijie is equivalent to giving the trick difference formula, which is more than 400 years earlier than the West.

The famous American historian of science Sutton commented that "Zhu Shijie was one of the most outstanding mathematicians of his time, and also throughout the ancient and modern worlds," and that The Four Elements of the Jade Annals is "one of the most important of the mathematical works of China, and at the same time one of the most outstanding mathematical works of the whole Middle Ages one of them." Zhu Shijie was not only an outstanding mathematician, but also a mathematics educator who traveled around the world and taught students for more than 20 years. He also personally compiled an introductory book on mathematics called The Enlightenment of Arithmetic. In the next volume of the Arithmetic Enlightenment, Zhu Shijie proposed the method of solving the hook shape with known hook chord sum and strand chord sum, which supplemented the deficiencies of the Nine Chapters of Arithmetic.