The development of mathematics is mainly based on the two basic concepts of number and shape, and the whole mathematics is developed around the refining, evolution and development of these two concepts. In the history of mathematical development, there are two parallel development routes of numbers and shapes, one is the arithmetic algebra route centered on development calculation, and the other is the geometric route centered on development shape. The former has two sources, one is the independent development of China mathematics, and the other is the ancient Babylonian number …
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Mathematics is an ancient discipline, which came into being with the appearance of human civilization, with a history of at least four or five thousand years. The original concepts and principles of mathematics sprouted in ancient times. After more than 4,000 years of joint efforts of many nationalities in the world, it has developed into such a huge system with rich content, many branches and wide applications. Understanding the development history of mathematics is helpful to cultivate students' interest in learning mathematics. The following contents hope to help them!
The development of mathematics is mainly based on the two basic concepts of number and shape, and the whole mathematics is developed around the refining, evolution and development of these two concepts. In the history of mathematical development, there have been two parallel development routes of numbers and shapes, one is arithmetic algebra with expanded calculation as the center, and the other is geometric route with expanded form as the main body. The former has two sources, one is the independent development of mathematics in China, and the other is Babylonian mathematics.
This route was further developed during the period of Alexandria in ancient Greece and carried forward in China, Indian and Arab countries. It was not until17th century that a complete elementary algebra was formed in Europe.
The route of "shape" is elementary geometry, which originated from Egyptian mathematics and made brilliant achievements in ancient Greece. These two kinds of mathematics merged in Europe in the17th century, and after further development, analytic geometry and variable mathematics appeared. Subsequently, due to the appearance of calculus, mathematics began a great change, resulting in a wide range of mathematical analysis fields, forming a three-legged situation of algebra, geometry and analysis.
In the 18 and 19 centuries, algebra, geometry and analysis formed their own different research fields due to the continuous differentiation of mathematics. The object of mathematical research is expanding day by day, and the concepts of number and shape are expanding and abstracting day by day, so that there is no trace of original calculation and simple graphics.
Geometry not only studies the spatial form of the material world, but also studies other forms and relationships similar to spatial forms and relationships, resulting in various new "spaces": Lobachevsky space, projective space, four-dimensional Riemannian space and various topological spaces, all of which have become the objects of geometry research. The objects of modern mathematics are common "quantities", such as vectors, matrices, tensors, spinors, hypercomplex numbers, groups, etc., and the operations of these quantities are studied.
These operations are similar to the four operations in arithmetic to some extent, but they are much more complicated. Vector is a simple example, and the addition of vectors is added according to the parallelogram rule. Abstract in modern algebra has reached the point that the term "quantity" has lost its meaning and has generally become a discussion of "things".
For such "objects", operations similar to ordinary algebraic operations can be performed. For example, two consecutive movements are equivalent to a total movement, and two algebraic transformations of the formula are equivalent to a total transformation.
Correspondingly, we can study the unique "addition" of a movement or transformation. Other similar operations are also studied in a wide range of abstract forms. The object of analysis has also developed greatly. In functional analysis, not only numbers are variables, but also functions themselves are variables.
The nature of a given function cannot be determined separately here, but depends on the relationship between the function and other functions. Therefore, what we see is not some individual functions, but a collection of all functions characterized by one way or another. This set of functions is combined into a "function space".
For example, consider the set of all curves on a plane or the set of all possible motions of a mechanical system, and determine the properties of a curve or motion on the basis of the relationship between a single curve or motion and other curves or motions. The common method in modern mathematics is to regard each function as a "point", the whole of a certain type of function as a "space", and the degree of difference between functions as a "distance" between "points", thus obtaining various infinite dimensional function spaces.
For example, the solution of a differential-integral equation group often boils down to the fixed point problem of geometric transformation in the corresponding function space, and the expansion of mathematical objects greatly expands the application scope of mathematics. Mathematical concepts have been widely introduced into physics, Einstein applied Riemann geometry to general relativity, von Neumann applied Hilbert space to quantum mechanics, Yang Zhenning and Mills applied fiber bundle theory to gauge fields, and so on.
From the second half of the19th century, that is, from Klein's "group" view to Cantor's establishment of set theory and axiomatic movement, the trend of mathematics towards integration became more and more obvious. The development of modern mathematics has promoted the deepening of the concepts of number and shape, and formed a variety of marginal disciplines. These disciplines not only did not deepen the separation between disciplines, but also led to the mutual connection and infiltration between disciplines, which made the previously basically separated fields communicate with each other and filled the gap between basic disciplines.
All disciplines have formed a solid organic whole. Marginal disciplines are not only produced in adjacent fields.
But also in far-off fields, the mutual penetration of basic disciplines has produced many comprehensive disciplines. The emergence and vigorous development of comprehensive disciplines indicates that the development of modern mathematics has shifted from the subject-led stage to the subject-led stage. The mutual penetration between disciplines is the embodiment of dialectics in which the two basic concepts of number and shape are closely linked.
The mathematization of various sciences has combined mathematics with other disciplines, resulting in many interdisciplinary disciplines, and many disciplines have derived many small branches, which not only promoted the development of various disciplines, but also enriched and developed the mathematics discipline itself.
However, no matter the division, combination, change and innovation of mathematics disciplines, no matter the internal changes of mathematics, although the territory of mathematics kingdom is constantly expanding, it is always controlled by the two basic concepts of number and shape. (Content is taken from * * * reading a book-History of Maritime Mathematics)
Abstract: In the early19th century, archaeologists excavated about 500,000 clay tablets carved with cuneiform characters, which spanned many periods of Babylonian history, and were densely engraved with strange symbols. After research, nearly 400 of them were identified as pure math tablets, including digital tables and some math problems. …
Editor's Note:/kloc-In the early 9th century, archaeologists unearthed about 500,000 clay tablets carved with cuneiform characters, which spanned many periods of Babylonian history, and were densely engraved with strange symbols. After research, nearly 400 of them were identified as pure math tablets, including digital tables and some math problems.
Archaeologists excavated about 500,000 clay tablets engraved with cuneiform characters in Mesopotamia in the first half of the19th century, which spanned many periods in Babylonian history. There are many strange symbols on these clay tablets. These symbols are actually the characters used by Babylonians, and people call them "cuneiform characters". Through research, scientists found that the clay tablets recorded the knowledge that Babylonians had acquired, and nearly 400 of them were identified as pure math boards, including digital tables and some math problems. Now the mathematical knowledge about Babylon comes from the analysis of these original documents.
arithmetic
The ancient Babylonians were skilled calculators, and their calculation programs were realized with the help of multiplication tables, reciprocal tables, square tables and cubic tables. The Babylonian method of writing numbers deserves our attention. They introduced a value system (hexadecimal) based on 60, which was still used by Greeks and Europeans for mathematical and astronomical calculations until16th century. Until now, hexadecimal is still used in angle, time and other records. For example, 1 m = 10 decimeter, 1 min =60 seconds and so on.
algebra
The ancient Babylonians had rich algebraic knowledge, and many clay tablets contained problems of linear equations and quadratic equations. Their process of solving quadratic equations is consistent with today's collocation method and formula method In addition, they also discussed some cubic equations and multivariate linear equations.
From BC 1900 to BC 1600, a table (Princeton No.322) was recorded, and it was found that there were two groups of numbers in it, namely the length of the hypotenuse of a right triangle and the length of a right angle, from which the length of another right angle was deduced, that is, the integer solution of the indefinite equation x2 y2=z2 was obtained.
geometry
The geometry of ancient Babylon is closely related to the actual measurement. They know that the corresponding edges of similar triangles are proportional, and they can calculate the area of a simple plane figure and a simple three-dimensional volume. We now divide the circumference into 360 equal parts because of the ancient Babylonians. The main feature of Babylonian geometry lies in its algebraic properties. For example, the problem of horizontal line parallel to one side of right triangle leads to quadratic equation; Cubic equation appeared when discussing the volume of prism.
The mathematical achievements of ancient Babylon reached a very high level in early civilization, but the accumulated knowledge was only the result of observation and experience, and there was no theoretical basis.
Abstract: Arithmetic and algebra are the most basic and oldest branches of mathematics, and they are closely related. Arithmetic is the foundation of algebra, and algebra evolved from arithmetic. The evolution from arithmetic to algebra is a major breakthrough in mathematical thinking methods.
Editor's Note The development of mathematics is not a simple accumulation of some new concepts, propositions and methods, but contains many fundamental changes in mathematics itself, that is, a qualitative leap. Several major breakthroughs in mathematical thinking methods in history have fully explained this point.
Arithmetic and algebra are the most basic and oldest branches of mathematics, and they are closely related. Arithmetic is the foundation of algebra, and algebra evolved from arithmetic. The evolution from arithmetic to algebra is a major breakthrough in mathematical thinking methods.
First, the historical inevitability of algebra.
Algebra is a research field of mathematics, and its initial and most basic branch is elementary algebra. The research object of elementary algebra is the operation of algebraic expressions and the solution of equations. Historically, elementary algebra is the continuation and popularization of arithmetic development. The contradiction of arithmetic's own movement and the need of social practice development provide the premise and foundation for the emergence of elementary algebra.
As we know, the main content of arithmetic is the nature of natural numbers, fractions and decimals and four operations. The appearance of arithmetic shows that human beings have taken a decisive first step in understanding the quantitative relationship in the real world. Arithmetic is an indispensable mathematical tool in the practical activities of human society, and it has a wide and important application in all departments of human society. Without this mathematical tool, the progress of science and technology is almost indistinguishable.
In the process of algorithm development, many new problems are raised due to the need of algorithm theory and practice development. One of the important problems is that the limitation of arithmetic problem-solving method limits the application scope of mathematics to a great extent.
The limitation of arithmetic problem-solving method mainly lies in that it is limited to the operation of specific known numbers, and abstract and unknown numbers are not allowed to participate in the operation. That is to say, when solving application problems by arithmetic, we should first collect and sort out all kinds of known data around the quantity we want, and list the formulas about these specific data according to the conditions of the problem, and then work out the results of the formulas through four operations of addition, subtraction, multiplication and division.
Many ancient mathematical application problems, such as trip problem, engineering problem, running water problem, distribution problem and profit and loss problem, were solved by this method. The key of arithmetic problem-solving method is to list arithmetic correctly, that is, to connect known data by adding, subtracting, multiplying and dividing symbols, and to establish a mathematical model that can reflect the essential characteristics of practical problems.
For those practical problems with simple quantitative relationship, it is not difficult to list the corresponding formulas, but for those practical problems with complex quantitative relationship, it is often difficult to list the corresponding formulas, and sometimes it requires high skills. Especially for those practical problems that contain several unknowns, sometimes they can't even be solved by establishing the formula of known numbers.
The limitation of arithmetic operation not only limits the application of mathematics, but also affects and restricts the continuous development of mathematics itself. With the in-depth development of mathematics itself and social practice, the limitations of arithmetic problem-solving methods are increasingly exposed, so the emergence of a new problem-solving method-algebraic problem-solving method has become a historical necessity.
The basic idea of algebraic problem solving method is: firstly, according to the conditions of the problem, an algebraic formula containing known numbers and unknowns is formed, and the equations are listed according to the equivalence relation, and then the values of the unknowns are obtained through the identity transformation of the equations. The central content of elementary algebra is to solve equations, so elementary algebra is usually understood as the science of solving equations.
The fundamental difference between elementary algebra and arithmetic is that the former allows unknowns to be the object of operation, while the latter excludes unknowns from operation. If an unknown number is also mentioned in arithmetic, then this unknown number can only play the role of symbolic equivalence of the operation result, and can only be located on the left side of the equation alone, and the formula on the right side of the static equation can complete the calculation of a specific number.
In other words, in arithmetic, the unknown has no right to participate in the operation. In algebra, as a conditional equation composed of known number and unknown number, the equation itself means that the known number and unknown number contained in it have the same operation state, that is, the unknown number has become the object of operation, and like the known number, it can participate in various operations and move from one side of multiplication to the other according to a certain law.
The process of solving the equation is essentially the process of transforming the unknown into the known number through the recombination of the known number and the unknown number, that is, putting the unknown on one side of the equation and the known number on the other side of the equation. In this sense, arithmetic operation is only a special case of algebraic operation, and it is the development and popularization of arithmetic operation.
Because of the universality and flexibility of algebraic operation, the generation of algebra has greatly expanded the application scope of mathematics, and many problems of arithmetic powerlessness can be easily solved in algebra. Moreover, the generation of algebra has had a great and far-reaching impact on the whole process of mathematics, and many important discoveries are related to the thinking method of algebra.
For example, the solution of quadratic equation leads to the discovery of imaginary number; The solution of quintic equation leads to the birth of group theory; The application of algebra in geometric problems leads to the establishment of analytic geometry and so on. Because of this, we regard the generation of algebra as the symbol of the first major turning point in mathematical thinking methods.
Second, the formation of algebraic architecture.
The original meaning of the word "algebra" is "the science of solving equations". So primitive algebra is also elementary algebra. As an independent branch of mathematics, elementary algebra has gone through a long historical process, and it is difficult for us to take a specific year as a sign of its appearance. Historically, it has gone through three different stages: word algebra, that is, expressing the object and process of operation in written language; Simplified word algebra, that is, simplified words are used to represent the contents and steps of operations; Symbolic algebra, that is, abstract letter symbols, is widely used.
The process from literal algebra to symbolic algebra is also the development process from immaturity to maturity of elementary algebra. In this process, Descartes, a French mathematician in the17th century, made outstanding contributions. He was the first person who advocated using x, y and z to represent unknowns. Many symbols he proposed and used are basically consistent with modern writing methods.
With the development of mathematics and the deepening of social practice, the research objects of algebra are constantly expanding, and the thinking methods are constantly innovating. Algebra develops from low-level form to advanced form, and from elementary algebra to advanced algebra. Higher algebra is rich in content and has many branches, the most basic of which are as follows.
Linear algebra: discuss the algebraic part of linear equation (linear equation), and its important tools are determinant and matrix.
Polynomial algebra: This paper mainly discusses the calculation and distribution of roots of algebraic equations with the help of the properties of polynomials, including divisibility theory, greatest common factor, factorization theorem, multiple factors and so on.
Group theory: a branch of algebra that studies the properties of groups and belongs to a field of abstract algebra. Group is an abstract algebraic system with operations. The concept of group theory was first put forward by the young French mathematician Galois at the beginning of19th century, and Galois became the founder of group theory. Today, group theory has been rich in content and widely used.
Ring theory: the branch of algebra that studies the properties of rings. It is a development field of abstract algebra. Ring is an abstract algebraic system with two operations and has many unique properties. A special ring is called a domain, and if the elements of a domain are numbers, it is called a number domain. Based on the concept of field, another field of abstract algebra-field theory is formed.
Boolean algebra: Also known as binary algebra, logic algebra or switch algebra, it is an abstract algebraic system with three operations. It was founded by British mathematician Bull in11940s. In recent decades, Boolean algebra has been widely used in circuit design, automation system and computer design.
In addition, there are branches such as lattice theory, Lie algebra and homology algebra.
There are great differences in thinking methods between advanced algebra and elementary algebra. Elementary algebra is computational, which is limited to the study of concrete number systems such as real numbers and complex numbers, while advanced algebra is conceptual and axiomatic, and its object is general abstract algebraic systems. Therefore, higher algebra is more abstract and universal than elementary algebra, which makes higher algebra more widely used. The development towards abstraction and universality is an important feature of modern algebra.
Cheng Xiaolong, the gold medal winner of the 40th International Maritime Organization (1999, Bucharest, Romania).
If you are familiar with high school mathematics, you will feel that it does not introduce many theories. Algebra is about the viewpoint of function and the nature of elementary function, the operation of trigonometric function, the principle of complex number and complex number vector, sequence and induction, and counting method. Analytic geometry introduces the method of describing geometric figures with quantitative language and the quantitative properties of several commonly used geometric figures. "Solid geometry" describes the position and measurement relationship of points, lines and surfaces in space, and focuses on several basic geometric bodies. To learn high school mathematics well, we must have an overall understanding and grasp of these knowledge, that is, to understand the role that the problems they solve play in mathematics and even practice.
Learning mathematics is by no means memorizing theorem formulas, nor is it empty problem-solving training. It is impossible to grasp the essence of mathematics only by paying attention to the formal surface of mathematics. The existence and development of mathematics is based on certain practical needs. Understanding this need, that is, the function of each part of mathematics, is helpful to understand mathematics as an organic whole, and it is difficult to truly understand mathematics without thinking. Therefore, it is particularly important to personally contact the mathematics in life.
This requires personal thinking and a thorough understanding of every problem in learning. I am usually used to analyzing and inferring the essence of a new concept by myself. When you encounter theorems and formulas, try to prove them yourself first, so that when you study the contents of books, you will have more experience and a deeper understanding of knowledge than you think.
For example, after doing this, why does a theorem have such restrictions, and it will be more clear if it is applicable in those circumstances. After understanding logical reasoning, we should go back and consider these conclusions as a whole and consider the dependence between the facts they describe and other mathematical knowledge. Doing so will also help to grasp knowledge from a macro perspective and have a deeper understanding of its main concepts. It is best to take some time to sort out this part of the theory and straighten out the relationship between its main knowledge points after learning some knowledge.
This is not a simple review, but to ensure that these things become your own knowledge. It is not simple reading, but deep thinking after understanding. Even you can abandon the textbook and complete this process only by thinking and necessary calculus, especially in the usual study, you only learn a small part of knowledge and do your homework at a time, which is relatively scattered. This overall familiarity is very necessary.
The necessary exercises can not only help you get familiar with what you have learned, but also help you understand the concepts and theorems you have learned and explore the deeper connotation of knowledge. Its other function, that is, the function of practice itself, is to exercise thinking. Thinking after completing the problem is of great benefit to the above two aspects, that is, doing the problem should not be limited to solving the problem itself, and sometimes you can think about the conclusions reflected by the problem and experience the methods and skills used. It is important to understand why this method is used, that is, you can understand the essence of the method.
When doing the problem, we must not ignore the reflection after pursuing too much, otherwise there will often be some unnecessary repetition, but it will not pay off. Another point is to think from different angles and not be satisfied with the existing methods, even if the existing methods are the simplest. Thinking and solving problems from another angle can bring some new gains, which will be more useful when doing more difficult problems.
Some people just write down all theorems and formulas, all kinds of problems and corresponding solutions. They may be able to cope with it when they have little knowledge, but once they have more content, it is difficult to sort it out. But it is relatively easy to master the basic thinking method of solving problems. The answer to a question may be very long, but the main idea of solving the problem may be only one or two, and most of the space is reasoning or operation.
Moreover, the way of thinking is connected with different parts of mathematics, and mastering it is the fundamental and the solution to all kinds of changes. The solution to the problem is by no means an unfounded inspiration, but a way that is produced after careful consideration in the process of solving the problem. Therefore, it is very important to understand this thinking process, that is, to see the essence through the phenomenon. Thinking method originates from the process of solving problems, and can only be mastered through independent thinking, analysis and exploration in the process of solving problems.
If one day, you find that you know all about the knowledge theory and thinking method in mathematics, then you can already master what you have learned well, plus some excellent basic skills, enough to cope with the general exam, but for a person who really wants to learn mathematics well, these are far from enough. As we all know, mathematics needs strict logical reasoning, but logical reasoning is not enough to represent the whole of mathematics.
As Courand, a great mathematician in this century, said, "It may be biased to overemphasize the mathematical characteristics of a formula. The core of mathematical theory is creative invention and intuitive elements that play a guiding and promoting role." Several important factors in mathematics are logic and intuition, analysis and creation, generality and individuality, and their comprehensive interaction constitutes the rich connotation of mathematics. To learn mathematics well, we must put ourselves in it, experience it by ourselves and discover it by ourselves.