1, real number and number axis. There is a one-to-one correspondence between real numbers and points on the number axis. By observing the characteristics of numbers on the number axis, it is intuitive and clear.

2. Application in solving equations (groups) or inequalities (groups). When solving equation problems with function images, the problem of equation roots is often regarded as the intersection of two function images. Solving inequality (group) by using number axis or function image is intuitive and vivid, and it is easy to find out the common parts of inequality (group) solutions or judge whether inequality groups have common solutions.

3. Application in function. It is a common method to study the properties of functions by using images. The geometric features and quantitative features of function images are closely combined, which embodies the characteristics and methods of combining numbers and shapes.

4. Application in geometry. For geometric problems, we often find out the quantitative relationship between edges and angles through graphs, and get the properties of graphs through the quantitative relationship between edges and angles.

Extended data:

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

The idea of combining numbers with shapes refers to a mathematical idea of using the properties of geometric figures to study the quantitative relationship to seek the solution of algebraic problems (using shapes to help numbers) or using the properties of geometric figures to solve geometric problems (using numbers to help shapes). The idea of combining numbers and shapes skillfully combines quantitative relations with geometric figures to solve problems.

The two basic objects of mathematical research "number" and "shape" number generally refer to written language, mathematical symbol language, algebraic expression and so on. Form, in layman's terms, generally refers to graphic language, functional image, algebraic geometric meaning and so on. Knowledge that can be expressed by both "number" and "shape" can be solved by the idea of combining number and shape. The thinking method of combining numbers and shapes is one of the main lines of mathematics teaching content, and it is applied.