1. The embryonic period of mathematics (before 600 BC);
2. The period of elementary mathematics (from 600 BC to1mid-7th century);
3. The period of variable mathematics (1mid-7th century to11920s);
4. Modern Mathematics Period (65438+11920s to World War II);
5. Modern Mathematics Period (since 1940s)
In the germination period of mathematics, after a long period of germination, mathematics has accumulated rich perceptual knowledge about numbers and shapes on the basis of its production. In the 6th century BC, the appearance of Greek geometry became the first turning point. Since then, mathematics has changed from a concrete and experimental stage to an abstract and theoretical stage, and elementary mathematics has been established. After continuous development and communication, it finally formed independent disciplines such as geometry, arithmetic, algebra and trigonometry. The oldest countries in the world are located in the big river basin: China in the Yellow River basin; Egypt in the lower Nile; Babylonia on the Euphrates and Tigris rivers; Indus and Ganges in India. These countries are all developed on the basis of agriculture, so we must master the laws of climate change in four seasons.
At present, the understanding of ancient Babylonian mathematics is mainly based on Babylonian clay tablets. These mathematical clay tablets show that Babylonia began to use the hexadecimal notation for more complicated calculations around 2000 BC, and there appeared a hexadecimal decimal, and its calculation rules were the same as those of integers. There are already tables about reciprocal, multiplication, square, cube, square root and cube root; With the help of the reciprocal table, division is often converted into multiplication for calculation. Babylonian mathematics has the characteristics of arithmetic and algebra, and geometry is only a way to express algebraic problems. At this time, there is no mathematical theory. The understanding of ancient Egyptian mathematics is mainly based on two volumes of cursive script. It can be seen from these two volumes of documents that ancient Egypt adopted the notation system of 10. Egyptians' mathematical interest lies in measuring land, while geometric problems are mostly measurement, involving the area of fields, the volume of barns and the simple calculation method of pyramids. However, because these calculation methods are conceived to solve the problems that must be solved in daily life such as land survey, grain distribution and capacity calculation after the Nile flood, there is no tendency to deduce formulas, theorems and proofs in theory. One of the main uses of Egyptian mathematics is astronomical research, which has also been developed. Due to geographical location and natural conditions, ancient Greece was influenced by ancient civilizations such as Egypt and Babylon, and became the earliest region in Europe to create civilization.
Greek mathematics is brilliant. The first period began in the 6th century BC and ended in the 4th century BC. Thales began the logical proof of propositions and the great development of mathematics in Greece. In the 5th century BC, Zhi Nuo of Elias School put forward four paradoxes about sports. Plato emphasized the important role of geometry in cultivating logical thinking ability, while Aristotle established formal logic and used it as a tool of proof. Democritus believed that geometric quantities were composed of many atoms that could not be subdivided. The second period was from the end of 4th century BC to the 1 century AD, and the academic center was transferred from Athens to Alexandria, so it was called the Alexandria period. During this period, many high-level mathematical manuscripts were produced, which have been passed down to this day. In the 3rd century BC, Euclid wrote the original works of plane geometry, proportion theory, number theory, irrational number theory and solid geometry, and established geometry on the deductive system for the first time, which became an epoch-making masterpiece in the history of mathematics and even thought. Later, Archimedes combined abstract mathematical theory with concrete engineering technology to explore the area and volume of geometric figures according to mechanical principles, which laid the foundation of calculus. Apolloni wrote the book "Conic Curve", which became the basis for later research on this issue. In the 1 century, Helen wrote books such as Measurement, explaining the quadrature method with concrete figures. Ptolemy in the second century A.D. completed the Masterpiece of Mathematical Astronomy at that time, Mathematical Compilation, and studied trigonometry in combination with astronomy. In the third century, Diophantine wrote arithmetic, which used abbreviations to solve indefinite equations and other problems. Its influence on the development of mathematics is second only to geometry. The three most outstanding achievements in Greek mathematics-Euclid's geometry, Archimedes' exhaustive method and Apolloni's conic theory-show that the main parts of mathematics at that time-arithmetic, algebra and geometry have been basically established.
The Romans conquered Greece and destroyed Greek culture. In 47 BC, the Romans burned down the Alexandria Library, and two and a half centuries' collection of books and 500,000 manuscripts were set on fire.
From the 5th century to15th century, the center of mathematics development shifted to India, Central Asia, Arab countries and China in the East. In this 1000 year, the rapid development of mathematics is mainly due to the needs of calculation, especially astronomy. Mathematics in ancient Greece attached importance to abstraction, logic and theory, and emphasized that mathematics was a tool for understanding nature, with emphasis on geometry. In ancient China and India, mathematics emphasized concreteness, experience and application, and mathematics was a tool to dominate nature, emphasizing arithmetic and algebra.
Indian mathematics is also an important part of world mathematics. Mathematics as a discipline has been established and developed. Indian mathematics is greatly influenced by Brahmanism, in addition to Greek, China and Near East mathematics, especially China.
In addition, Arabic mathematics also plays an important role. Arabs improved the counting system in India, and the research object of "algebra" was defined as equation theory. Let geometry be subordinate to algebra and not attach importance to proof; By introducing trigonometric functions such as tangent, cotangent, secant and cotangent, an accurate trigonometric function table is made, and some important formulas of plane triangle and spherical triangle are found, which makes trigonometry independent from astronomy.
In China, calculation has been widely used in the Spring and Autumn Period and the Warring States Period, and decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world. During this period, econometrics was widely used in production, and mathematics was improved accordingly. A hundred schools of thought contended during the Warring States Period also promoted the development of mathematics. Qin and Han dynasties were the rising period of feudal society, with rapid economic and cultural development. The ancient mathematical system of China was formed in this period, and its main symbol was that arithmetic became a specialized subject, and the emergence of mathematical works represented by Nine Chapters of Arithmetic.
Nine Chapters Arithmetic is a summary of the development of mathematics during the establishment and consolidation of feudal society in the Warring States, Qin and Han Dynasties. As far as its mathematical achievements are concerned, it is a world-famous mathematical work. The work of Zhao Shuang and Liu Hui in Wei and Jin Dynasties laid the theoretical foundation of China's ancient mathematical system. Liu Hui proved by infinite division that the volume ratio of right-angled square cone to right-angled tetrahedron is always 2: 1, which solved the key problem of general solid volume. When proving the volume of square cone, cylinder, cone and frustum, Liu Hui put forward the correct method to solve the volume of sphere completely. Since then, China's mathematics has been further developed under the impetus of mathematicians such as Qin Jiushao, Zu Chongzhi, Guo Shoujing and Cheng Dawei.
In the history of western Europe, the darkness of the Middle Ages hindered the development of mathematics to some extent. The Renaissance in Europe began in the15th century and further developed European mathematics. /kloc-mathematical activities in the 0/5th century focused on arithmetic, algebra and trigonometry. Miao Lei's masterpiece Encyclopedia of Triangle is the first systematic exposition of plane and sphere by Europeans, which is independent of astronomy. /kloc-in the 6th century, Tattaglia discovered the algebraic solution of cubic equation, accepted negative numbers and used imaginary numbers. /kloc-The greatest mathematician in the 6th century was David, who wrote many works on trigonometry, algebra and geometry, among which the most famous Introduction to Analytical Methods improved symbols and greatly changed algebra. Steven created decimals. /kloc-At the beginning of the 7th century, the invention of logarithm was a great achievement of elementary mathematics. 16 14 years, Naipur pioneered logarithm, 1624 years, Briggs introduced logarithm, which is equivalent to the common logarithm now, thus making the calculation method a big step forward. At this point, the main parts of elementary mathematics-arithmetic, algebra and geometry have been formed and matured.
The mathematical period of variables is from1mid-7th century to11920s. The main contents of mathematical research in this period are quantitative change and geometric transformation. The main achievements of this period are analytic geometry, calculus, advanced algebra and other disciplines.
17th century is a groundbreaking century. Three great events of great significance to mathematics have happened in this century. The first is the appearance of Galileo's experimental mathematical method, which shows the new combination of mathematics and natural science. Its characteristic is to find out some measurable factors in the studied phenomena and study the changing law of these quantities by mathematical methods. The second important event was the publication of Descartes' important book On Method and its appendix Geometry in 1637. It introduces the concepts of moving point coordinates, variables and functions. Because of the coordinates, the relationship between plane curve and binary equation is established, which leads to a new discipline-analytic geometry, which studies geometry by algebraic method. This is a turning point in mathematics and the first decisive step in the development of variable mathematics. The third important event was the establishment of calculus, and the most important work was independently completed by Newton and Leibniz. They realized that differential and integral are actually a pair of inverse operations, and thus gave the basic theorem of calculus, namely Newton-Leibniz formula. /kloc-many profound and obvious changes have taken place in mathematics in the 0/7th century. In the scope of mathematics activities, mathematics education has expanded, the number of people engaged in mathematics work has increased rapidly, mathematics works have been widely spread, and various associations have been established. In the traditional aspect of mathematics, from the study of form to the study of logarithm, algebra occupies a dominant position. In the development trend of mathematics, the process of scientific mathematization has begun. Mathematicization of mechanics first appeared, represented by Newton's Mathematical Principles of Natural Philosophy written in 1687. Starting from the three laws and through mathematical logic reasoning, the laws of mechanics are inevitably extended one by one. 18th century, trigonometry, analytic geometry, calculus, number theory, equation theory and other mathematical disciplines developed rapidly. 19 A great mathematical achievement appeared in the 1920s, which firmly established the theoretical foundation of calculus on the concept of limit. Cauchy developed the acceptable limit theory in the book 182 1 Analysis Course, and then defined the continuity, derivative and integral of functions very strictly, emphasized the necessity of studying the convergence of series, and gave the root discrimination and integral discrimination of positive series. During this period, the appearance of non-Euclidean geometry became a major event in the history of mathematics, which changed people's view that only Euclidean geometry existed. His revolutionary thought not only paved the way for new geometry, but also was the prelude and preparation for the emergence of the theory of relativity in the 20th century. At this time, people found the correct geometry-non-Euclidean geometry, which is different from the usual Euclidean geometry. The ideological emancipation caused by non-Euclidean geometry is of great significance to modern mathematics and science, because human beings have finally begun to break through the limitations of senses and go deep into nature. Riemann sum and Lobachevsky made great contributions to the discovery of non-Euclidean geometry. Riemann popularized the concept of space and created a broader field of geometry-Riemann geometry. Later, Hamilton discovered an algebra-quaternion algebra, in which the multiplicative commutative law did not hold. The appearance of noncommutative algebra has changed people's view that it is unthinkable to have an algebra different from ordinary arithmetic algebra. His revolutionary ideas opened the door to modern algebra. On the other hand, the concept of group is introduced because of the exploration of the conditions for finding the roots of unary equations. From the 1920s to 1930s, Abel and Galois initiated the study of modern algebra. At this time, the research object of algebra expanded to vectors, matrices and so on, and gradually turned to the study of algebraic system structure itself. /kloc-in the 0/9th century, the third far-reaching mathematical event occurred: the arithmeticization of analysis. In 1874, Wilstrass put forward a famous idea called "Arithmetic of Analysis". First, the real number system itself should be strictly defined, and then all the concepts of analysis should be deduced from this number system. /kloc-In the late 20th century, due to the work of Dedekind, Cantor and piano, these mathematical foundations have been established on a simpler and more basic natural number system.
From the 1940s to 1950s, three earth-shattering events occurred in the history of world science, namely, the utilization of atomic energy, the invention of electronic computers and the rise of space technology. In addition, many new situations have emerged, which have caused drastic changes in mathematics. 1945 after the birth of the first electronic computer, a huge science naturally formed around it because of its wide application and great influence. The appearance of computers has promoted the development of mathematics, which is divided into three fields: pure mathematics, computer mathematics and applied mathematics. Although modern mathematics presents a colorful situation, its main characteristics can be summarized as follows: (1) The object and content of mathematics have developed greatly in depth and breadth, the ideas, theories and methods of analysis, algebra and geometry have changed greatly, and the trend of continuous differentiation and synthesis of mathematics is strengthening. (2) The entry of electronic computers into the field of mathematics has had a great and far-reaching impact. (3) Mathematics has penetrated into almost all scientific fields and is playing an increasingly important role. Pure mathematics has been developing in depth, and mathematical logic and mathematical foundation have become the foundation of the whole mathematics building.