Lagrange's most outstanding contribution to mathematics is to separate mathematical analysis from geometry and mechanics, which makes the independence of mathematics more clear. Since then, mathematics is no longer just a tool for other disciplines.

There are mainly the following four aspects:

Equation solution

In the first ten years of working in Berlin, Lagrange spent a lot of time in solving algebraic equations and transcendental equations, made valuable contributions and promoted the development of algebra. He submitted two famous papers to the Berlin Academy of Sciences: On Solving Numerical Equations and Research on Algebraic Solutions of Equations. He summed up the various solutions of cubic and quartic algebraic equations by predecessors into a set of standard methods, that is, turning the Lagrangian point of the equation into a low-order equation (called

permutation group

He tried to find the resolvent function of the quintic equation, hoping that this function would be the solution of the equation below quintic, but failed. However, the concept of permutation group has been included in his thought, which has inspired Abel and Galois later, and finally solved the problem why the general equation with quartic degree or more cannot be solved by algebraic method. So Lagrange can be said to be the pioneer of group theory.

number theory

In number theory, Lagrange also showed extraordinary talent. He answered many questions raised by Fermat, such as the question that a positive integer is not more than the sum of four squares and so on. He also proved the irrationality of pi. Joseph Lagrange's research results enrich the content of number theory.

power series

In analytic function theory and his paper as early as 1772, he made a unique attempt, which laid a theoretical foundation for calculus. He tried to simplify the differential operation into algebraic operation, thus abandoning the infinitesimal that was confusing since Newton, and wanted to establish all analytical studies. But because he didn't consider the convergence of infinite series, he thought he got rid of the concept of limit, but in fact he just avoided it. He failed to achieve the goal of algebraic and rigorous calculus. However, his method of expressing functions by power series had an impact on the development of analysis and became the starting point of the theory of real variable functions.