Numbers and shapes are the two oldest and most basic research objects in mathematics, and they can be transformed into each other under certain conditions. The object of middle school mathematics research can be divided into two parts: number and shape. There is a connection between numbers and shapes, which is called the combination of numbers and shapes, or the combination of numbers and shapes As a mathematical thinking method, the application of the combination of numbers and shapes can be roughly divided into two situations: either by means of the accuracy of numbers to clarify some properties of shapes, or by means of the geometric intuition of shapes to clarify some relationship between numbers, that is, the combination of numbers and shapes includes two aspects: the first situation is "solving shapes with numbers" and the second situation is "helping numbers with shapes". "Solving shapes by numbers" means that some shapes are too simple to see any laws by direct observation, and it is necessary to assign values to the shapes, such as side length and angle.

Hua, a famous mathematician in China, once said: "The combination of numbers and shapes is good in all aspects, but everything is wrong when it is separated." "Number" and "shape" reflect two attributes of things. We think that the combination of numbers and shapes is mainly exponential one-to-one correspondence. The combination of numbers and shapes is to combine abstract mathematical language and quantitative relationship with intuitive geometric figures and positional relationships. Through the combination of abstract thinking and image thinking, complex problems are simplified and abstract problems are concretized, thus optimizing the way to solve problems.

The combination of numbers and shapes is a common thinking method in solving mathematical problems. The combination of numbers and shapes can make some abstract mathematical problems intuitive and vivid, change abstract thinking into image thinking, and help to grasp the essence of mathematical problems. In addition, due to the combination of numbers and shapes, many problems are easy to solve and the solutions are simple.

2. The so-called combination of numbers and shapes is the idea of solving mathematical problems through the mutual transformation of numbers and shapes according to the corresponding relationship between numbers and shapes, which is often related to the following contents: (1) the corresponding relationship between real numbers and points on the number axis; (2) correspondence between function and image; (3) The correspondence between curve and equation; (4) Concepts based on geometric elements and geometric conditions, such as complex numbers and trigonometric functions; (5) The structure of a given equation or algebraic expression has obvious geometric significance. Such as equation.