Because there is no biography of Jia Xian in the history books, we can't know the life story of this mathematician today. I only know that he used to be a small official in the Song Dynasty, a student of the astronomical mathematician Chu Yan at that time, and wrote two mathematical works, but now they are all lost. Fortunately, Yang Hui, a mathematician in the Southern Song Dynasty, quoted many materials of Jia Xian's mathematical thought in his book, which made us understand Jia Xian's great contribution to today's mathematics.
Jia Xian's most famous mathematical achievement is that he created a digital schema, that is, "the root diagram of radical method" (see the figure below). This picture is now found in Yang Hui's book, but after quoting this picture, Yang Hui specifically stated: "Jia Xian used this technique." Therefore, in the past, the mathematical circles in China called this picture "Yang Hui Triangle", which was actually inappropriate, but should be called "Jia Xiansan"
Angle "is the most appropriate.
Graph 1-6- 1 cholesky decomposition root graph
In modern mathematical terms, this "root graph of radical method" is actually a positive integer binomial theorem coefficient table. Readers who know a little bit about algebra know:
If we arrange the coefficients to the right of the equal sign in the above formula, we can get:
This is completely consistent with the figures in the root map of root practice.
This binomial coefficient expansion law is called "Pascal Triangle" in the history of western mathematics. Pascal, a French mathematician, gave a numerical triangle table in his book 1654, which is similar to Jia Xian's Root Graph of Radical Method (see figure 1-6- 1). In fact, in Europe, a similar number triangle was not first invented by Pascal, but it did not spread widely at first. The oldest digital triangle in the west can be traced back to1527; But compared with this map of Jia Xian, it is more than 400 years late. Therefore, we have every reason to call this mathematical achievement first invented by China "Jia Xian Triangle" and go down in history.
Not only that, Jia Xian's diagram also contains the law of numbers in the diagram. Careful readers may have found that both hypotenuses of this triangle are composed of the number 1, while other numbers are equal to the sum of the two numbers on its shoulder. According to this rule, this digital triangle can be written as any number of layers; In other words, binomial is any positive integer power.
According to this diagram, the coefficient expansion can be easily obtained.
Figure 1-6-2 Pascal triangle
According to Yang Hui's records, Jia Xian's method of finding all the coefficients in the Root Making Method is the new method he used in root making and root making. By applying this "multiply-multiply-open method", we can not only get the expansion coefficient of any high order, but also get the root of any high order. Before Jia Xian, it was more than 1000 years from Han Dynasty to Tang Dynasty.
At one time, ancient mathematicians in China could only perform square root and square root operations of positive numbers, and there was no good method for square roots of higher powers above the fourth power. It was not until the publication of Jia Xian's "Method of Multiplication and Opening" that the best opening method of higher power was really found, which can be used to open the higher power of any rational number. This is very valuable in the history of Chinese mathematics and even in the history of world mathematics. Later mathematicians continued to advance on this basis and extended it to numerical solutions of arbitrary higher-order equations. Qin, a mathematician in the Southern Song Dynasty, finally developed the numerical solution of higher-order equations with increase, multiplication and expansion as the main body to a very complete degree. In Qin's works, the coefficients of the equation are both positive and negative; There are both integers and decimals; The degree of the equation is as high as 10. For example:
Its solution is basically consistent with Horner's method (given by British mathematician Horner in 18 19), but it is more than 500 years earlier than Horner's method. The numerical solution of higher order equations is an outstanding creation of China's mathematics in Song and Yuan Dynasties, which developed gradually from Jia Xian to Qin Dynasty.