1. Function idea: use a function to express a mathematical problem and use a function to explore the general law of this problem. This is the most basic and commonly used mathematical method. 2. The idea of combining numbers with shapes: Algebra is combined with geometry, for example, algebraic method is used to solve geometric problems, and geometric method is used to solve algebraic problems, which is the most commonly used method in analytic geometry. For example, find the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((a-1) 2+b.3. The idea of classified discussion: when a problem may lead to different results because of different situations of a certain quantity, it is necessary to discuss the various situations of this quantity in categories. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A.4.. Equation Thought: When a problem may be related to an equation, we can solve the problem by constructing the equation and studying its properties. For example, when proving Cauchy inequality, we can transform Cauchy inequality into a discriminant of quadratic equation.