Mathematics (mathematics; Greek: μ α θ η μ α κ? In the west, this word comes from the ancient Greek word μ? θξμα(máthēma) has learning, learning and science, and it has another narrow and technical meaning-"mathematical research", even in its etymology. Its adjective meaning is related to study or hard work, and it can also be used to refer to mathematics. Its surface plural form in English and its surface plural form in French, les mathématiques, can be traced back to the neutral plural mathematica in Latin, which is Cicero's name from the Greek plural τ α α θ η μ α ι κ? (ta mathēmatiká), a Greek word used by Aristotle, refers to the concept of "everything counts". (Latin: Mathemetica) means counting and counting technology. In ancient China, mathematics was called arithmetic, also called arithmetic, and finally changed to mathematics. If you want to learn math well, you must practice hard.

History of mathematics

The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress until the Renaissance in16th century, and the mathematical innovation generated by the interaction with new scientific discoveries led to the acceleration of knowledge, until today. Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, and sometimes it will lead to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics, that is, mathematics itself, without any practical application. Although many studies started from pure mathematics, many applications will be found later. The French Bourbaki School, founded in 1930s, believes that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. Boone School believes that there are three basic abstract structures: algebraic structure (group, ring, domain …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …).

In this field of editing mathematics research.

The main subjects of mathematics mainly come from the needs of commercial calculation, understanding the relationship between numbers, measuring land and predicting astronomical events. These four requirements are generally related to a wide range of sub-fields in mathematics, such as quantity, structure, space and change (that is, arithmetic, algebra, geometry and analysis). In addition to the above-mentioned main concerns, there are sub-fields used to explore the relationship between the core of mathematics and other fields: to logic, to set theory (foundation), to empirical mathematics in different sciences (applied mathematics), and to the rigorous study of uncertainty in modern times. The study of quantity and quantity begins with numbers, and at the beginning is the familiar arithmetic operations of natural numbers and integers and those described in arithmetic. In number theory, the deeper properties of integers are studied, including famous results, such as Fermat's last theorem. Number theory also includes two unsolved problems that are widely discussed: twin prime conjecture and Goldbach conjecture. When the number system is further developed, integers are considered as a subset of rational numbers and included in real numbers, and continuous quantities are represented by real numbers. Real numbers can be further extended to complex numbers. Further generalization of numbers can continue to include quaternions and octal numbers. The consideration of natural numbers will also lead to over-limit numbers, which formulizes the concept of counting to infinity. Another research field is its size, which leads to cardinality and another infinite concept: Avery number, which allows meaningful comparison between the sizes of infinite sets. Structure Many mathematical objects, such as sets of numbers and functions, have embedded structures. The structural properties of these objects are discussed in groups, rings, bodies and other abstract systems that are themselves objects. This is the field of abstract algebra. Here is a very important concept, that is, vector, which is extended to vector space and studied in linear algebra. The study of vector combines three basic fields of mathematics: quantity, structure and space. Vector analysis extends it to the fourth basic field, namely change. The study of space comes from geometry, especially Euclidean geometry. Trigonometry combines space and numbers, including the famous Pythagorean theorem. Nowadays, the study of space extends to higher dimensional geometry, non-Euclidean geometry (which plays a central role in general relativity) and topology. Numbers and spaces play an important role in analytic geometry, differential geometry and algebraic geometry. In differential geometry, there are concepts such as fiber bundle and calculation on manifold. Algebraic geometry has the description of geometric objects such as polynomial equation solution set, which combines the concepts of number and space; There is also the study of topological groups, which combines structure and space. Lie groups are used to study space, structure and change. Among its many branches, topology is probably the most advanced field in mathematics in the twentieth century, and it includes the long-standing Poincare conjecture and the controversial four-color theorem, which have only been proved by computers and never verified by manpower. In order to clarify the mathematical foundation, mathematical logic and set theory, the foundation and philosophy were developed. German mathematician Georg Cantor (1845- 19 18) initiated the set theory, boldly marched into infinity, provided a solid foundation for all branches of mathematics, and its own content was quite rich, put forward the existence of real infinity, and made inestimable contributions to the future development of mathematics. Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Even the mathematician Pi Aucar, who is famous for his "profound and creative", compared set theory to an interesting "morbid situation", and even his teacher Kroneck hit back at Cantor as a "mental derangement" and "walked into a hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said: "My theory is as firm as a rock, and anyone who opposes it will shoot himself in the foot." He also pointed out: "The essence of mathematics lies in its freedom, and it is not bound by traditional ideas." This argument lasted for ten years. Cantor suffered from schizophrenia at 1884 because of frequent depression, and finally died in a mental hospital. However, after all, history has fairly evaluated his creation. Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. At the beginning of the 20th century, Hilbert, the greatest mathematician in the world, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Russell praised Cantor's works as "the greatest works that can be boasted in this era". Mathematical logic focuses on putting mathematics on a solid axiomatic framework and studying the results of this framework. It is the birthplace of Godel's second incomplete theorem, which is perhaps the most widely spread achievement in logic-there is always a true theorem that cannot be proved. Modern logic is divided into recursion theory, model theory and proof theory, which are closely related to theoretical computer science. Engels said: "Mathematics is a science that studies the quantitative relationship and spatial form of the existing world."

Edit the mathematical classification in this paragraph.

Discrete mathematics fuzzy mathematics

Five branches of mathematics

1. Classical Mathematics II. Modern mathematics iii. Computer mathematics. Random mathematics 5. Economic mathematics

Branch of mathematics

1 .arithmetic 2. Elementary algebra 3. Advanced algebra iv. Number theory 5. Euclidean geometry 6. Non-Euclidean geometry 7. Analytic geometry 8. Differential geometry 9. Algebraic geometry 10. Projective geometry 1 1. Geometric topology 12. Topology 65438. 3. Fractal geometry 14. Calculus 15. Theory of real variable function 16. Probability statistics 17. Complex variable function theory 18. Functional analysis 19. Partial differential equation 20. Ordinary differential equation 2 1. Mathematical logic.

Generalized mathematical classification

Vertical division: 1. Elementary Mathematics and Ancient Mathematics: This refers to the mathematics before17th century. Mainly Euclidean geometry established by ancient Greece, arithmetic established by ancient China, ancient Indian and ancient Babylon, algebraic equations developed during the European Renaissance, etc. 2. Variable mathematics: refers to mathematics established and developed at the beginning of 17- 19 century. The period of variable mathematics from the first half of17th century can be divided into two stages: the establishment stage of17th century (heroic age) and the development stage of18th century (creative age). 3. Modern mathematics: refers to19th century mathematics. The19th century of modern mathematics is the stage of comprehensive development and maturity of mathematics, and the face of mathematics has undergone profound changes. Most branches of mathematics were formed in this period, and the whole mathematics showed a comprehensive prosperity. 4. Modern mathematics: refers to mathematics in the 20th century. 1900, the famous German mathematician D. Hilbert delivered a famous speech at the World Congress of Mathematicians, and put forward 23 mathematical problems (see below) to predict and understand the future development of mathematics, which opened the prelude of modern mathematics in the 20th century. 1900, at the International Congress of Mathematicians held in Paris, Hilbert delivered a famous speech entitled "Mathematical Problems". According to the achievements and development trend of mathematical research in the past, especially in the 19th century, he put forward 23 most important mathematical problems. These 23 problems, collectively called Hilbert problems, later became the difficulties that many mathematicians tried to overcome, which had a far-reaching impact on the research and development of modern mathematics and played a positive role in promoting it. Some Hilbert problems have been satisfactorily solved, while others have not yet been solved. The belief that every mathematical problem can be solved in his speech is a great encouragement to mathematicians. Hilbert's 23 problems belong to four blocks: 1 to 6 are basic mathematical problems; Questions 7 to 12 are number theory problems; Problems 13 to 18 belong to algebraic and geometric problems; 19 to 23 belong to mathematical analysis. Now only one list is listed: (1) Cardinality problem of Cantor continuum. (2) Arithmetic axiom system is not contradictory. (3) It is impossible to prove that two tetrahedrons with equal base and equal height are equal in volume only according to the contract axiom. (4) Take a straight line as the shortest distance between two points. (5) Conditions for topology to be a Lie group (topological group). (6) Axiomatization of physics, which plays an important role in mathematics. (7) Proof of transcendence of some numbers. (8) The distribution of prime numbers, especially for Riemann conjecture, Goldbach conjecture and twin prime numbers. (9) Proof of the general law of reciprocity in arbitrary number field. (10) Can we judge whether an indefinite equation has a rational integer solution by finite steps? Quadratic theory in (1 1) algebraic number field. Composition of (12) class domain. The impossibility of (13) combination of binary continuous functions to solve the seventh general algebraic equation. The finite proof of (14) some complete function systems. (15) Establish the foundation of algebraic geometry. Topological research on (16) algebraic curves and surfaces. The square sum representation of (17) semi-positive definite form. (18) Construct space with congruent polyhedron. (19) Is the solution of the regular variational problem always an analytic function? (20) Study the general boundary value problem. (2 1) Proof of the existence of solutions for Fuchs-like linear differential equations with given singularities and single-valued groups. (22) Automorphic single-valued analytic function. (23) Carry out the research of variational method. Horizontal division: 1. Pure mathematics. Also known as pure mathematics or pure mathematics, it is the core part of mathematics, including algebra, geometry and analysis, which study numbers, shapes and the relationship between numbers and shapes respectively. 2. Applied mathematics. Simply put, it is the application of mathematics. 3. Computational mathematics. Learn calculation methods (numerical analysis), mathematical logic, symbolic mathematics, computational complexity, programming and other issues. This subject is closely related to computers. 4. Probability and mathematical statistics. Divided into probability theory and mathematical statistics. 5. Operational research and control. Operational research is a subject that solves the operation, organization and management problems of complex systems such as manpower, material resources and financial resources by mathematical methods on the basis of establishing models.

Edit the symbols, language and rigor of this paragraph.

In modern symbols, simple expressions can describe complex concepts. This image is generated by a simple equation. Most of the mathematical symbols we use today were invented after16th century. Before that, mathematics was written in words, which was a hard procedure that would limit the development of mathematics. Today's symbols make mathematics easier to be controlled by experts, but beginners are often afraid of it. It is extremely compressed: several symbols contain a lot of information. Like music notation, today's mathematical symbols have clear grammar and information codes, so it is difficult to write them in other ways. Mathematical language is also difficult for beginners. How to make these words have more accurate meanings than everyday language? Novices are also troubled. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. But these special symbols and terms are used for a reason: mathematics needs accuracy more than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor". Stiffness is a very important and basic part of mathematical proof. Mathematicians hope that their reasoning and axioms of the definite reason system can be inferred. This is to avoid the wrong "theorem", relying on unreliable intuition, there have been many examples in history. The rigor expected in mathematics changes with time: the Greeks expected careful argumentation, but in Newton's time, the methods used were not so rigorous. Newton's definition of solving problems was not carefully analyzed and formally proved until the19th century. Today, mathematicians have been arguing about the rigor of computer-aided proof. When a large number of measurements are difficult to verify, it is hard to say that they are effective and rigorous.

Edit the development history of mathematics in this section.

Mathematics, the history of world mathematics development, originated from early human production activities. It is one of the six great arts in ancient China, and it is also regarded as the starting point of philosophy by ancient Greek scholars. Mathematical Greek μ α θ η μ α κ? Mathematickó s) means "the basis of learning" and comes from μαρθξμα(máthema) ("science, knowledge and learning"). The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric people also knew how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge of geometry at that time. In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history. Since the historical era, the main principles in mathematics have been formed, which are used for tax and trade calculation, for understanding the relationship between numbers, for measuring land and predicting astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics. By16th century, elementary mathematics, such as arithmetic, elementary algebra and trigonometry, had been basically completed. The appearance of the concept of variables in the17th century made people begin to study the relationship between variables and the mutual transformation between graphs. In the process of studying classical mechanics, the method of calculus was invented. With the further development of natural science and technology, set theory and mathematical logic, which are produced for studying the basis of mathematics, have also begun to develop slowly.