In ancient China, solving equations was called "Fang Zi". During the Song and Yuan Dynasties, prescriptions developed to a new historical stage and reached the advanced level in the world at that time.
China has a long history in ancient times, especially in mathematics. There are many interesting math problems circulating among the people, which are generally expressed by catchy poems. There are many equation problems. For example, there is a poem asking Zhou Yu's age:
Dajiangdong has gone to the waves, and there are many people throughout the ages. In the 1930s, the governor of Wu Dong died young and at a double-digit rate. Ten is smaller than one, and one is six times the symbol of longevity. Which student is faster? How many years did he belong to Zhou Yu?
According to the meaning of the question, Zhou Yu's age is double digits, and the unit number is 3 larger than the ten digits. If the decimal digit is x, the unit number is (x+3). From the equation of "6 times the number of units and longevity symbol", it is 6 (x+3) = 10x+(x+3) and the solution is x=3, so these ancient equations are very interesting, which not only popularizes mathematical knowledge, but also stimulates people's mathematical thinking.
In ancient mathematics, sequence equation and solution equation are two important interrelated problems. Before the Song Dynasty, mathematicians had to list an equation, such as The Classic of Ancient Calculations written by Wang Xiaotong, a famous mathematician in the Tang Dynasty, which proposed the solution of the positive root of cubic equation for the first time, which could solve the calculation problem of different widths in engineering construction. It is an outstanding contribution to China's ancient mathematical theory, more than 300 years earlier than Arabs and more than 600 years earlier than Europe.
With the development of mathematical research in the Song Dynasty, a perfect method of solving equations appeared, which directly promoted the research on the method of sequence equations, and then another outstanding creation of China's mathematics, Tianshu, appeared.
According to historical records, there were a number of works about celestial arts in Jin and Yuan Dynasties, especially the works of mathematicians Ye Li and Zhu Shijie, which clearly expounded celestial arts.
In her mathematical monograph "Measuring the Round Sea Mirror", Ye Li comprehensively discussed the steps, skills, algorithms and symbolic representations of establishing unknowns and equations through the Pythagorean inclusion problem, which made celestial art develop to a quite mature new stage.
Yi Gu Yan Duan is a concise and easy-to-learn introductory book written by Ye Li for Tianyuan beginners. He is the author of Jing Zhai Gu Hu Jin, Jing Zhai Anthology, Bi Shu Cong Jian, Pan Shuo and so on. The former has a collection of 12 volumes, while the last three volumes have been lost.
Zhu Shijie's "Arithmetic Enlightenment", which includes commonly used data, conversion between weights and measures and field area units, four calculation algorithms, calculation simplification, fraction, proportion, area, volume, surplus and deficiency, higher-order arithmetic progression summation, solution of numerical equations, solution of linear equations, celestial theory, etc., is a relatively comprehensive mathematical enlightenment book.
Zhu Shijie's masterpiece "Siyuan Encounter" records the establishment and solution of the higher-order equation he created, as well as his important achievements in higher-order arithmetic progression summation and higher-order interpolation.
In addition to Ye Li and Zhu Shijie, the General Theory of River Defense written by Yuan Semu Shan Si also has the application of natural science in water conservancy projects.
Astrology is a general method of using unknown equations, which is basically the same as the method of equations in algebra now, but the writing is different. It must first "set Tianyuan as a certain", which is equivalent to "set X as a certain", and then list two equal algebraic expressions according to the given conditions of the problem. Then through a similar process of merging similar terms, an equation with one end zero is obtained.
The emergence of celestial sphere provides a unified method for equations, and its steps are much more advanced than the algebra of Arab mathematicians. In Europe, this was not done until16th century.
After the celestial sphere technique, mathematicians quickly extended this method to higher-order multivariate equations, and finally Zhu Shijie founded the quaternary technique.
Since the simultaneous equations of multivariate linear equations were put forward in Nine Chapters Arithmetic, there has been no significant progress for centuries.
In the aspect of sequence equation, Jiangzhou's subsection method has prepared for the art of heaven, and has the idea of finding equivalent polynomials; Dong He is a pioneer of celestial science, but their equation derivation is still bound by geometric thinking. Ye Li basically got rid of this bondage and summed up a set of fixed celestial surgery procedures, which made celestial surgery enter a mature stage.
Jia Xian gave the method of increasing, multiplying and opening when solving the equation, while Liu Yi found the positive root of the quartic equation with the positive and negative square roots. On this basis, Qin solved the numerical problem of higher-order equations.
So far, the establishment and solution of one-dimensional higher-order equations have been realized.
Linear equations have existed since ancient times, so they have the conditions for generating multivariate higher power equations. Li Dezai's binary technique and Liu Dajian's ternary technique appeared one after another. Zhu Shijie summarized and improved binary technology and ternary technology, developed "celestial technology" into "quaternary technology" and established the theory of quaternary higher order equation.
Zhu Shijie, an outstanding mathematician in Yuan Dynasty, illustrated the arrangement methods and solutions of one-dimensional, two-dimensional, three-dimensional and four-dimensional equations with examples. Some examples are quite complicated and the numbers are staggering. Not only have I never seen it before, but it is also rare today. It can be seen that Zhu Shijie has mastered the solution of multivariate higher order equations very skillfully.
"Quaternary technology" is a method to establish and solve higher-order multivariate equations. Solving an equation with quaternion is to put all the coefficients of the equation in a square matrix.
Among them, the word "Tai" is still recorded on the right side of the constant term, the coefficients of four unknown first-order terms are placed at the top and bottom of the constant term, and the coefficients of higher-order terms are expanded outward one by one according to the power, and the intersection points of rows and columns represent the products of the corresponding unknown powers respectively.
When solving this equation group represented by a square matrix, we should use the elimination method to gradually transform the equation into a one-dimensional higher-order equation, and then find the positive root by adding, multiplying and opening.
Judging from the representation of quaternion, this square matrix form is not only difficult to calculate, but also difficult to represent equations with more than four unknowns, which has great limitations.
China's algebra reached its peak in the period of Quaternary technology. If we want to go further, we need to find another way. Later, the progress of algebra in Qing dynasty was realized by the in-depth study of equation theory by Wang Lai and others and the introduction of western mathematics.