1, rational numbers and irrational numbers are collectively called real numbers. 2. There is a one-to-one correspondence between real numbers and points on the number axis. On the number axis, the number represented by the right point is greater than that represented by the left point. 3. In the range of real numbers, antonyms, reciprocal and absolute values have exactly the same meanings as rational numbers. Real numbers can be added, subtracted and absolute values. Moreover, the arithmetic and arithmetic laws of rational numbers still apply to real numbers. Real number theory For thousands of years, mathematics enthusiasts have been making unremitting efforts to find a reliable logical basis for the whole mathematics, but the arithmeticization of analysis is based on real numbers. If we don't understand the essence of real numbers, give a clear definition of real numbers, and establish the theory of real number size and operation, the properties of continuous functions can't be fully understood, and even the sufficiency of Cauchy convergence criterion can't be strictly proved. This forces mathematicians to speed up the pace of establishing mathematical theories. The core problem of real number theory is the understanding of irrational numbers. As early as the beginning of19th century, Cauchy felt the importance of defining irrational numbers. In the process of analysis, he defined irrational number as the limit of convergent rational number series, and set {yn} as rational number series. If there is a number y, yn-> Y, then y is an irrational number. This definition has logical defects. Because the rational number sequence {yn} does not converge to irrational number (that is, y is rational number), it is impossible to define irrational number; If you don't converge to a rational number, you have to admit that y is an irrational number before you can define it as nothing, which makes the mistake of circular definition. Since the late 1960s, there have been several different definitions of irrational numbers, which were founded by Villtras, Meire, Cantor and Dai Dejin. But no matter how different their specific methods of defining real numbers are, they all meet the following three conditions: first, regard irrational numbers as known, and define irrational numbers from rational numbers; Secondly, the defined LV and its operation law are one-third of rational numbers, so the real number defined in this way is complete, that is, no new number will appear under the limit operation. In order to avoid mistakes in Cauchy's definition of rational number, Wilstrass insisted on his own position and introduced the concept of "composite number". Rational numbers are defined by composite numbers. For example, 3(2/3) consists of 3α and 2β, where α= 1 is the main unit and β= 1/3. When you know which elements a number consists of and how many times each element appears, the number is completely determined. Wilstrass then defined irrational numbers such as √2 as 1α, 4 β1γ-the irrational numbers defined by Cantor and Mill are basically the same, and introduced a new kind of numbers-real numbers based on rational numbers. This number category includes rational numbers and irrational numbers. Among Lu's theoretical achievements, Dai Dejin's real number theory is the most complete. The idea that people define real numbers by dividing rational numbers comes from the consideration of straight line continuity. Almost at the same time, Ren and Cantor put forward the hypothesis that the set of real numbers corresponds to the points on a straight line one by one. This assumption was later called "Cantor-Dai Dejin" axiom. He thinks that the rational points on a straight line are discontinuous, and the space must be filled with infinite numbers to make the straight line continuous. How do we express the irrational numbers of these vacancies? Dai Dejin used the division of all rational numbers to represent an irrational number. The above definitions of irrational numbers regard rational numbers as known, because any rational number can be written as the ratio of two integers, so the problem boils down to integers. So do you need to redefine integers? There are also differences on this issue, and Wilstrass thinks it is unnecessary. Rational numbers are logically classified as a pair of integers, so there is no need to further study the logic of integers. In his book The Nature and Meaning of Numbers, Dai Dejin gave an integer theory with the idea of set theory. Although it was not adopted because it was too complicated, it gave piano a direct inspiration. 1889, Italian mathematician piano gave the theory of natural numbers by axiomatic method in his book A New Method of Arithmetic Principles, thus completing the logicalization of the whole number system. Piano was born in Turin. He is a lecturer and professor at the University of Turin and a mathematical logician. Unlike logicians, he advocates that mathematics should be based on logic, but logic should be used as a mathematical tool. In his book Arithmetic Principles and Methods, piano used a series of symbols, such as ∑, no and a+, to represent the class of natural numbers and the next natural number of A, etc. Four undefined original concepts are given: set, natural number, successor number and attribution; Five axioms of natural numbers are also put forward: 1) 1 is a natural number; 2) 1 is not the successor of any natural number; 3) Every natural number A is not a successor number A+; 4) if a+=b+, then a = b；; 5) If S is a set of natural numbers containing 1, if S contains both A and a+, then S contains all natural numbers. This axiom is the logical basis of mathematical induction. Then, piano defined integers according to natural numbers: Let A and B be natural numbers. Then the number pair (a,) or "a-b" defines an integer. When a>b, a/span > With the concept of integer, rational number is defined by ordered pair: if both n and m are integers, ordered pair (n, m) (M0), that is, n/m, defines a rational number. In this way, piano established the natural number system, integer system and rational number system concisely on the basis of natural number axiom by using mathematical symbols and axiom methods. Of course, the number system constructed by axioms and logical methods makes a mathematician feel unnatural. They think this is an incomprehensible extension of this clear concept. However, the establishment of real number theory has written a brilliant chapter in the history of mathematics in19th century.