Before  BC, the first numbers that human ancestors knew were natural numbers.

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Since then, with the development of human civilization, the scope of numbers has continued to expand. This expansion is related to the needs of social practice on the one hand and the need for number operations on the other. Here we only explain the operation requirements of numbers.

　　For example, within the range of natural numbers, the difference between two natural numbers is not necessarily a natural number. To make the natural numbers closed to subtraction, negative numbers were introduced. In this way, human understanding of numbers expanded from natural numbers to integersset of numbers in which integers are also closed for division operations led to the extension of the set of integers to the set of rational numbers .

　　Any rational number can be expressed as

( where , is an integer, and ) ,

Compared with integers, rational numbers have good properties that integers do not have.

　　For example, any two rational numbers contain an infinite number of rational numbers, which is the so-called density of the rational number set ; another example is that any rational number can find a unique corresponding point on the number axis ( called a rational point) . On the number axis, the rational points are arranged in order of size from right to left. This is the so-called orderliness：of the rational number set .

　　Although the rational points are dense on the number line, they do not fill the entire number line. For example, for a square with side length x , that its diagonal length is (see the figure below) , then according to the Pythagorean theorem , we have

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Although this point definitely falls on the number axis, there is no rational point corresponding to it on the number axis. This shows that there are many gaps besides rational points on the number axis. It also shows that although rational numbers are very dense, they are not There is no continuity. We call the points at these gaps irrational points, and the numbers corresponding to the irrational points are called irrational numbers . Irrational numbers are infinite non-repeating decimals, such as , , etc.

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　　All rational numbers and irrational numbers are called real numbers , which extends the set of rational numbers to the set of real numbers . The set of real numbersthe four arithmetic operations , but also for the square root operation. It can be proved that real number points can cover the entire number axis without leaving any gaps. This is the so-called continuity of real numbers century that mathematicians fully understood real numbers and their related theories .

　　Since any real number is given, there is a unique point corresponding to it on the number axis; conversely, any point on the number axis also corresponds to a unique real number. It can be seen that the set of real numbers is equivalent to the set of points on the entire number axiscommonly used real number sets learned in middle school : the set of natural numbers is denoted as , the set of integers is denoted as , the set of rational numbers is denoted as , and the set of real numbers is denoted as . Between these number sets The relationship is as follows:

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