After a long period of brewing, the ideas and methods of calculus bred in ancient Greece finally emerged under the stimulus of industrial revolution at the initiative of Newton and Leibniz in the17th century. The birth of calculus creatively pushed mathematics to a new height. It declared the basic end of classical mathematics and marked the beginning of modern mathematics with variables as the research object. Although the concept of early calculus is still rough and its reliability is still in doubt, its outstanding power in computing technology dwarfs all previous traditional mathematics. Through the invention of calculus, people saw the new happiness of mathematics. Throughout the seventeenth and eighteenth centuries, almost all European mathematicians showed great interest and positive dedication to calculus. Criticism of tradition, pursuit of new methods and expansion of new fields have enabled them to compose a "heroic symphony" in the history of mathematics! As the contemporary analyst R.Courant pointed out: "Calculus ... this subject is the crystallization of a shocking intellectual struggle; This struggle has gone through more than 2500 years, and it is deeply rooted in many fields of human activities. Moreover, as long as people strive to know themselves and nature, this struggle will continue. "Analytical Arithmetic is a vivid manifestation of this struggle. The work of Porzano, Cauchy, Westeras and others provided rigor for the analysis. These works liberated calculus and its popularization from the complete dependence on geometric concepts, movements and intuitive understanding. These studies caused a great sensation from the beginning. It is said that Cauchy put forward the theory of series convergence at a scientific conference of the Paris Academy of Sciences. After the meeting, Laplace hurried home so as not to miss anyone, and checked the series he used in "Celestial Mechanics". Fortunately, every series used in the book is convergent. The rigor of analysis promotes the understanding that the lack of clear understanding of the digital system itself must be compensated. For example, a key mistake in Porzano's proof of "Zero Theorem" of closed interval continuous function is that he lacks sufficient understanding of real number system; For the in-depth study of limit, we also need to know the real number system. Cauchy can't prove the sufficiency of his convergence criterion of sequence, which is also due to his lack of in-depth understanding of the structure of real number system. Wilstras pointed out that in order to establish the properties of continuous functions in detail, the theory of arithmetic continuum is needed, which is the fundamental basis of analytical arithmeticization. 1872 is the most memorable year in the history of modern mathematics. This year, F.Kline put forward the famous Erlanger scheme, and Weierstrass gave a famous example of a function that is continuous everywhere but differentiable everywhere. Also in this year, three theories of real numbers were put forward: Dedekind's "segmentation" theory; Cantor, Heine and Meire's "basic sequence" theory and Wilstrass's "bounded monotone sequence" theory appeared in Germany at the same time. The purpose of trying to establish real numbers is to give a formal logical definition, which does not depend on the meaning of geometry and avoids the logical error of defining irrational numbers with limits. With these definitions as the basis, there will be no theoretical cycle in the derivation of the basic theorem of limit in calculus. Derivative and integral can therefore be directly based on these definitions, without any properties associated with perceptual knowledge. The concept of geometry cannot be fully understood and accurate, which has been proved in the long years of the development of calculus. Therefore, the necessary strictness can only be achieved through the concept of number and after cutting off the relationship between the concepts of number and geometric quantity. Here, Dedekind's work is highly praised, because the real number defined by Dedekind's division is the intuitive creation of human wisdom completely independent of space and time. In 1858, Dedekind expressed his desire to find a way to make analysis strict when teaching calculus. He said, "... never think that it is scientific to introduce differential calculus in this way. This is recognized. As for myself, I can't restrain this dissatisfaction and make up my mind to study this problem until I establish a purely arithmetic and completely strict foundation for the principle of infinitesimal analysis. "Dedekind doesn't consider how to define irrational numbers, so as to avoid a vicious circle of Cauchy. But consider what exists in the continuous geometric quantity when the arithmetic method obviously fails: what is the essence of continuity? Thinking along this direction, Dedekind understands that the continuity of a straight line cannot be explained by fuzzy clustering, but can only be regarded as the nature of dividing a straight line by points. He saw that the points on a straight line were divided into two categories, so that every point in one category was on the left of every point in the other category, and there was one point and only one point, which resulted in this kind of cutting. This is not true for an ordered rational number system. This is why the points on a straight line form a continuum, and rational numbers are impossible. As Dai Dejin said, "From such a common point of view, the secret of continuity is exposed. "In essence, the theory of three schools of real numbers gives a strict definition of irrational numbers, thus establishing a complete real number field. The successful construction of real number field has completely bridged the gap between arithmetic and geometry for more than two thousand years, and irrational numbers are no longer "irrational numbers". The ancient Greeks' idea of arithmetic continuum was finally realized in a strict scientific sense. The next goal is to give the definition and properties of rational numbers. Ohm, Wilstrass, Kroneck and Peano have done outstanding work in this field. 1859, Weierstrass and others realized that as long as natural numbers are known, there is no need for further axioms to establish real numbers. Therefore, the key to establishing real number theory is rational number system, and the core of establishing rational number system lies in constructing the basis of ordinary integers and establishing the properties of integers. Between 65438 and 0872-78, Dedekind gave an integer theory. 1889, Peano introduced an integer with a set of axioms for the first time, thus establishing a complete theory of natural numbers. The symbols created by Peano, such as "∈" for belonging, ""for inclusion, N0 for natural number category, and a+ for the next natural number after A, still have far-reaching influence today. But who can believe that it is because he used these symbols in class that the students rebelled. He tried to satisfy them by passing all the exams, but it was useless. So he was forced to resign as a professor at the University of Turin. Kroneck said, "God created integers, and everything else is man-made" (the rest is man's work). (Reference [5], p477) However, in the process of arithmetic analysis, integers are not exempted because they are the darling of God. Seeking unity is an important driving force for the development of mathematics. Looking back at the whole process of "analysis arithmeticization", we find that people don't know where the end point is at the starting point and how to go. From the Pythagorean school's discovery of the measure of incommensurability to the concern for the concept of infinity caused by Zeno's paradox, various studies leading to calculus were born. When Dedekind, Cantor, Weierstrass and others established irrational numbers on the basis of rational numbers, and finally Peano gave the logical axiom of natural numbers, the theory of rational numbers was finally completed, so the basic problems of real number system were finally announced. The basic concepts of calculus-the limit of continuous variables: derivative and integral, are as rigorous as Euclidean geometry in logic and form. China sages have an old saying: 99 is one! If we understand the "one" here as "1", which is the first natural number, then Pythagoras' famous saying about the historical development of calculus is astounding: everything has a number! (It's all numbers. ) 1900, at the second international mathematics conference held in Paris, Poincare proudly praised: "If we take pains to analyze it strictly today, we will find that it is impossible to deceive us only by relying on syllogism or intuition attributed to pure numbers. Today, we can say that we have reached absolute strictness. "