Mathematics is a science that studies quantitative relations and spatial forms in the real world. Simply put, it is the science of studying numbers and shapes. Due to the needs of life and labor, even the most primitive people know simple counting, and it has developed from counting with fingers or objects to counting with numbers. In China, at the latest in Shang Dynasty, there was a method of expressing large numbers by decimals. By the Qin and Han Dynasties, there had been a perfect decimal system. Nine Chapters Arithmetic, which is not later than the first century A.D., contains the calculation rules of square roots and square roots that are only possible with numerical system, as well as various operations of fractions and solutions of linear simultaneous equations, and introduces the concept of negative numbers. In his annotated Nine Chapters Arithmetic, Liu Hui also proposed to use decimals to represent odd square roots of irrational numbers. But it was not until the Tang and Song Dynasties (in Europe, after Steven in16th century) that decimals began to be popularized. In this book, Liu Hui used the circumference of a circle inscribed by a regular polygon to approximate the circumference of a circle, which became a common method for later generations to find pi. Although China never had a general concept of irrational numbers or real numbers, in essence, China had completed all the operation rules and methods of the real number system at that time, which was not only indispensable in application. It is also indispensable for the early education of mathematics. As for Europe, which has inherited the culture of Babylon, Egypt and Greece, it focuses on the study of the nature of numbers and the logical relationship between these properties. As early as Euclid's Elements of Geometry, there were some conclusions such as the concept of prime number, the infinite number of prime numbers and the unique decomposition of integers. Numbers without fractions were discovered in ancient Greece, and now they are called irrational numbers.16th century. In modern times, the concept of numbers was further abstracted. According to the different operation rules of numbers, the general number system was discussed independently in theory, forming several different branches of mathematics. Square root and square root are necessary operations to solve the simplest equation of higher order. In Nine Chapters Arithmetic, a special form of quadratic equation is solved. In the Song and Yuan Dynasties, a clear concept of "Tianyuan" (i.e. unknown number) was introduced, and methods for finding numerical solutions of higher-order equations and simultaneous equations of higher-order algebra with up to four unknowns appeared, commonly known as Tianyuan method and quaternary method. The expressions, algorithms and elimination methods of related polynomials are close to modern algebra. Outside China, ramiz's works in the ninth century in Arabia expounded the solution of quadratic equation, which is generally regarded as the originator of algebra. Its solution has essentially the same style as the geometric method relying on cutting in ancient China. China's ancient mathematics devoted itself to the concrete solution of equations, while the traditional European mathematics originated from ancient Greece and Egypt was different. /kloc-in the 6th century, David replaced the coefficients of equations with words and introduced algebraic symbolic calculus. Discuss the properties of algebraic equations, which are determinant, matrix, linear space and linear transformation derived from linear equations. From the introduction of concepts such as complex numbers and symmetric functions caused by algebraic equations to the establishment of Galois theory and group theory, the extremely active algebraic geometry in modern times is nothing more than a theoretical study of the set formed by the solutions of high-order simultaneous algebraic equations. The study of shape belongs to the category of geometry. The ancient people all had a simple concept of shape, which was often expressed by pictures, and the reason why graphics became mathematical objects was due to the requirements of making and measuring tools. In ancient China, Yu Xia had some measuring tools, such as ruler, moment, standard and rope. In Mo Jing, a series of geometric concepts were abstractly summarized and scientifically defined. Zhou Pian and Liu Hui's Shu Jing Dao give the general method and concrete formula of observing heaven and earth with moments. In Liu Hui's Nine Chapters Arithmetic and Nine Chapters Arithmetic, the principle of two to one (Liu Hui's principle) requires the volume of polyhedron; In the 5th century, in order to find the volume of a curve, especially the volume of a sphere, Zu (Riheng) put forward the principle that "if the potentials are the same, the products cannot be different". There is also the limit method (secant method) of approaching the circumference with inscribed regular polygons. However, since the Five Dynasties (about 10 century), China has made little achievements in geometry. The central task of China's geometry is to measure and calculate the measurement of area and volume, while the ancient Greek tradition attaches importance to the relationship between the properties of a shape and various properties. Euclid's Elements of Geometry established definitions, axioms and theorems. It has become a model of axiomatization in modern mathematics and influenced the development of the whole mathematics. Especially, the study of parallel axioms led to the appearance of non-Euclidean geometry in19th century. In Europe, since the Renaissance, through the study of the perspective relationship of painting, projective geometry has emerged. /kloc-in the 8th century, gaspard monge applied analytical methods to study shapes, which was the first in differential geometry. Gauss's surface theory and Riemann's manifold theory create the independence of shape and surrounding space. /kloc-In the 9th century, Klein unified geometry from the point of view of groups. In addition, Cantor's point set theory, for example, expands the range of shapes. Poincare founded topology, making the continuity of shape the object of geometric research. All these make geometry look brand new. In the real world, numbers and shapes are inseparable. China's ancient mathematics reflected this objective reality, and numbers and shapes have always been complementary and developed in parallel. For example, Pythagoras survey puts forward the requirement of square root, and both square root and square root methods are based on the consideration of geometry. Most of them also come from geometric and practical problems. In Song and Yuan Dynasties, due to the introduction of Tianyuan concept and polynomial equivalence concept, geometric algebra appeared. In the catalogue and map drawing of astronomical geography, numbers have been used to represent places, but they have not developed to the point of coordinate geometry. In Europe, in the14th century, Ao' Erslmu's works have sprouted, about the graphic representation and function of latitude and longitude. /kloc-in the 0/7th century, Descartes put forward a systematic method to represent geometric things by algebra and its application. Under its enlightenment, through the work of Leibniz and Newton, it developed into a modern form of coordinate analytic geometry, which made the unification of numbers and shapes more perfect, not only changed the old geometric proof method that followed Euclid geometry in the past, but also caused the generation of derivatives. It has become the root of calculus. This is a great event in the history of mathematics. In the 17th century, due to the requirements of science and technology, mathematicians were urged to study movement and change, including the change of quantity and the transformation of shape (such as projection), and the concepts of function and infinitesimal analysis, which is now calculus, were also produced, which made mathematics enter a new era of studying variables. Since the 18th century, with the establishment of analytic geometry and calculus as an opportunity, mathematics has developed rapidly on an unprecedented scale, and many branches have appeared. Because most of the objective laws of nature are expressed in the form of differential equations, the study of differential equations has received great attention from the beginning. Differential geometry was born at the same time as calculus, and the work of Gauss and Riemann produced modern differential geometry. At the beginning of the 20th century, Poincare founded topology. It opens a way for the qualitative and overall study of continuous phenomena. The analysis of random phenomena in the objective world produces probability theory. The military needs of the Second World War and the complexity of large-scale industry and management have produced operations research, system theory, cybernetics, mathematical statistics and other disciplines. Practical problems need specific numerical solutions. Computational mathematics came into being. The requirement of choosing the best way has produced various optimization theories and methods. The development of mechanics, physics and mathematics always influences and promotes each other, especially relativity and quantum mechanics, which promote the development of differential geometry and functional analysis. In addition, in the19th century, equation chemistry was used only once, and some cutting-edge mathematical knowledge has been used by creatures with little connection with mathematics. /kloc-In the late 9th century, the emergence of set theory also entered a critical era, which promoted the formation and development of mathematical logic, and also produced various ideological trends and basic schools of mathematics that regarded mathematics as a whole. Especially in 1900, the German mathematician Hilbert gave a speech on important issues of contemporary mathematics at the second international congress of mathematicians, and the rise of the French Bourbaki school, which developed in the 1930s and viewed mathematics as a whole with a structural concept. It had a great and far-reaching influence on the development of mathematics in the 20th century, and the word "scientific mathematization" began to be enjoyed by people. The edge of mathematics continues to penetrate and expand into natural science, engineering technology and even social science, and some marginal mathematics has appeared. The inherent demand of mathematics itself has also spawned many new theories and branches. At the same time, its core part is constantly consolidated and improved, and sometimes it is properly adjusted to meet external needs. In a word, the tree of mathematics is flourishing and has deep roots. In the process of the vigorous development of mathematics, the concepts of number and shape are constantly expanding and becoming more and more abstract, so that there is no trace of initial counting and simple graphics. Nevertheless, there are still some objects and operational relationships expressed in geometric terms in the new branch of mathematics. For example, think of a function as a point in a certain space. This method is effective. In the final analysis, it is because mathematicians have been familiar with the simple relationship between mathematical operations and graphics, and have a long-term and profound practical foundation. Moreover, even the most primitive numbers such as 1, 2, 3, 4, and geometric figures such as points and straight lines are highly abstract concepts. Therefore, if number and shape are understood as generalized abstract concepts, the aforementioned mathematics, as the scientific definition of studying number and shape, can also be applied to modern mathematics at this stage. Because the quantitative relationship and spatial form of mathematical research objects come from the real world, although mathematics is highly abstract in form, it is always rooted in the real world. Life practice and technical requirements will always be the real source of mathematics. Conversely, mathematics plays an important and key role in the practice of transforming the world. The enrichment, perfection and wide application of theory have always been accompanied and promoted each other in the history of mathematics. However, due to the different objective conditions of different nationalities and regions, the specific development process of mathematics is also different. Generally speaking, the ancient Chinese nation used bamboo as the fund for fund-raising and operation. Naturally, it leads to the decimal value system. The superiority of the calculation method is helpful to the concrete solution of practical problems. The mathematics developed from this has formed a unique system characterized by constructiveness, computability, programmability and mechanization. The main goal is to start from the problem and then solve it. In ancient Greece, thinking was emphasized. Pursuing the understanding of the universe, it has developed into an axiomatic deductive system with abstract mathematical concepts and properties and their logical interdependence as the research objects. After China's mathematical system reached its peak in the Song and Yuan Dynasties, it stagnated and almost disappeared. In Europe, a series of changes, such as the Renaissance, the religious revolution and the bourgeois revolution, led to the industrial revolution and the technological revolution. The use of machines has a long history at home and abroad. But in China, in the early Ming Dynasty, it was dismissed as a strange skill by the emperor. In Europe, it developed because of the development of industry and commerce and the stimulation of navigation. Machines liberate people from heavy manual labor and lead them to theoretical mechanics and general scientific research on movement and change. Mathematicians at that time actively participated in these changes and the solution of corresponding mathematical problems, and produced positive results. The birth of analytic geometry and calculus has become a turning point in the development of mathematics. /kloc-the leap in mathematics since the 0/7th century can generally be regarded as the continuation and development of these achievements. In the 20th century, all kinds of brand-new technologies appeared, resulting in a new technological revolution, especially the appearance of computers. Mathematics is facing a new era. A feature of this era is the gradual mechanization of some mental work. Different from the dominant ideas and methods of mathematics since17th century, discrete mathematics and group mathematics have attracted attention because of the development and application of computers. The role of computer in mathematics is not limited to numerical calculation. The importance of symbolic operation is becoming more and more obvious (including mathematical research such as machine proof). Computers are also widely used in scientific experiments. In order to better cooperate with computers, the requirements of mathematics for constructiveness, computability, programmability and mechanization are also quite prominent. Algebraic geometry is a highly abstract mathematics, and the recent formulation of calculating algebraic geometry and constructing algebraic geometry is one of its clues. In a word, mathematics is developing with the new technological revolution.