mathmatics . . . . .

What is mathematics?

Mathematics is a science that studies the relationship between spatial form and quantity in the real world. It is divided into elementary mathematics and advanced mathematics. It is widely used in scientific development and modern life production, and is an essential basic tool for studying and studying modern science and technology.

Introduction to mathematical symbols

In a word, mathematics is an infinite science.

2. The characteristics of mathematics

strict

Mathematical language is also difficult for beginners. How to make these words have more accurate meanings than everyday language also puzzles beginners. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. However, these special symbols and proper nouns are used for a reason: mathematics needs accuracy more than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor".

Stiffness is an important and basic part of mathematical proof. Mathematicians want their theorems to be derived from axioms through systematic reasoning. This is to avoid wrong "theorems" and unreliable intuition, and many examples have appeared in history. The expected rigor 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 dealt with again until the19th century through careful analysis and formal proof. Today, mathematicians are arguing about the rigor of computer-aided proof. When a large number of measurements are difficult to verify, their proofs can hardly be said to be effective and rigorous. Because of the differences of the times, a lot of knowledge has been erased, but mathematics will never be erased, and wisdom will always be circulated.

3. The application of mathematics

Life is inseparable from mathematics, and mathematics is inseparable from life. Mathematical knowledge comes from life and is higher than life, and ultimately serves life. Indeed, learning mathematics should be applied in real life. Mathematics is used by people to solve practical problems, in fact, mathematical problems arise from life. For example, there are countless problems such as addition, subtraction, multiplication and division when shopping on the street, and floor plan when building a house. This kind of knowledge is produced in life. In mathematics teaching, we should give students the opportunity to practice, guide students to consciously use mathematical knowledge, analyze and solve practical problems in life with mathematical knowledge and methods, make life problems mathematical, and let students deeply appreciate the application value of mathematics.

Curriculum standards emphasize the process of abstracting practical problems into mathematical models, explaining and applying them from students' existing life experience. In fact, most of the teaching contents of primary school mathematics can be related to students' real life. Teachers should find the "meeting point" between the content of each class and students' real life, and stimulate students' interest in learning mathematics and enthusiasm for participating in learning. In teaching, the teacher's responsibility is not only to induce students' desire to solve practical problems, but also to let students learn to choose the necessary conditions and information from many conditions and information to solve real-life problems and experience the success and happiness of applying mathematics to solve practical problems.

First, solve the problems in life and apply what you have learned.

The new curriculum standard points out that students should "realize that there is a lot of mathematical information in real life." Mathematics is widely used in the real world. In the face of practical problems, we can actively try to use the knowledge and methods we have learned from the perspective of mathematics to seek strategies to solve problems ... "... We often encounter this situation, and students still don't understand a topic for a long time. If the teacher links this problem with real life, the students can solve it immediately. Therefore, as teachers, we should think about how to make full use of students' existing life experience and guide students to apply mathematics knowledge to reality, so as to realize the application value of mathematics in life.

Second, the creation of life scenes to stimulate interest in learning

Application problems originate from life, and each application problem can always find its own blueprint in life. Therefore, in the teaching of applied problems, if applied problems are combined with real life, students' interest in learning can be stimulated.

Thirdly, restore the essence of life and cultivate students' thinking.

While paying attention to mathematics life, every teacher must fully realize that the essence of mathematics teaching is to develop students' thinking. Life does not mean simplification of mathematical knowledge. On the contrary, returning mathematics to the essence of life is more conducive to the development of students' thinking.

I once saw a report that a professor asked a group of foreign students, "How many times did the minute hand and the hour hand overlap between 12 and 1?" Those students all took off their watches from their wrists and began to wave; When the professor tells the same question to the students in China, the students will use mathematical formulas to make calculations. The commentary said that it can be seen that China students' mathematical knowledge is transferred from books to their brains, so they can't use it flexibly. They seldom think of learning, applying and mastering mathematics knowledge in real life.

Fourth, realize the needs of life and promote the development of the subject.

From the perspective of educational psychology, life has five different levels of needs, and the highest needs are the needs of self-realization and decision-making. Once we link the teaching and life of application problems in teaching, the potential needs of students will be stronger.

Five. The importance of mathematics

Take famous sayings as evidence:

Everything has been calculated-Pythagoras

In the mathematical world, what matters is not what we know, but how we know it. Pythagoras

Beauty of Mathematical Symbols

Numbers rule the universe. Pythagoras

Geometry has no king. -Euclid

I'm determined to give up just abstract geometry. In other words, I don't think about the problems that are only used to practice my thoughts. I did this to study another kind of geometry, that is, geometry aimed at explaining natural phenomena. -Descartes (René 1596- 1650)

Mathematics is the most powerful knowledge tool left by human knowledge activities and the root of some phenomena. Mathematics is immutable and exists objectively, and God will build the universe according to the laws of mathematics. Descartes

Imaginary number is a wonderful human spiritual sustenance, which seems to be an amphibian between being and not being. -gottfried William von Leibniz 1646- 17 16)

Things that don't work don't exist. -Leibniz

After considering several things, the whole thing comes down to pure geometry, which is a goal of physics and mechanics. -Leibniz

Although we are not allowed to see through the secrets of nature, so as to know the real reason of the phenomenon, some fictitious assumptions may still occur to explain many phenomena. —— leonhard euler (1707- 1783)

Because the structure of the universe is God's most perfect and wise creation, nothing will happen if there is no certain maximum or minimum law in the universe. -Euler

Some beautiful theorems in mathematics have the following characteristics: they are easy to be summarized from facts, but their proofs are extremely hidden. Mathematics is the king of science. Gauss

Mathematics is the head of natural science, and number theory is the queen of mathematics. Gauss

This is the advantage of a well-structured language, and its simplified symbols are usually the source of abstruse theories. -Laplace (Pierre-Simon Laplace 1749- 1827)

In mathematical science, the main tools for us to find truth are induction and analogy. Laplace

Read Euler, read Euler, he is our teacher. Laplace

Only by vigorously developing mathematics can a country show its strong national strength. Laplace

Understanding a giant's research method is as important to scientific progress as discovering itself. Scientific research methods are often the most interesting part. Laplace

It would be a serious mistake to think that it is only necessary in geometric proof or sensory proof. -augustin louis cauchy (1789- 1857)

Paper full of mathematical formulas

Give me five coefficients and I will draw an elephant; Give me the sixth coefficient, and the elephant will wag its tail. -Cauchy

If mankind is adding many new terms to science and letting readers continue to study the wonderful and difficult things before them, then he must be sure that science has made great progress. -Cauchy

Geometry sometimes seems that essentials precede analysis, but in fact geometry precedes analysis, just like a servant walking in front of his master, opening the way for him. -james joseph sylvester (18 14- 1897)

Maybe I can claim the title of Adam in mathematics without undue demands, because I believe that the creation of mathematical rationality is named by me (which has become popular and universal) more than all other mathematicians of the same age combined. Sylvester

A mathematician who is not a poet will never be a complete mathematician. -Karl Weierstrass (18 15- 1897)

The essence of mathematics lies in freedom. -William Conrad

In the field of mathematics, the art of asking questions is more important than the art of answering questions. -Cantor

As long as a branch of science can raise a large number of questions, it is full of vitality, and no questions indicate the termination or decline of independent development. -Hilbert

Music can stimulate or soothe feelings, painting can make people pleasing to the eye, poetry can touch people's hearts, philosophy can make people gain wisdom, science can improve material life, but mathematics can give them all. Klein

No subject can clarify the harmony of nature more clearly than mathematics. Paul carus

Problem is the core of mathematics-P.R. halmos.

Where there are numbers, there is beauty! -Plocque Lars.

Logic is invincible because it must be opposed. -Boutros

Mathematical subsystem is as vast as nature itself-Fourier

Logic can wait, because it is eternal-Havisham.

Only by successfully applying mathematics can a science be truly perfect. Marx

Mathematics is an infinite science. Herman Weil

History makes people wise, poetry makes people witty, and mathematics makes people careful. bacon

The scientific level of a country can be measured by the mathematics it consumes. -Rao

No subject can clarify the harmony of nature more clearly than mathematics. Carlos

Mathematics is the judge and master of law and theory. Benjamin

Mathematics and culture of intransitive verbs

The cultural value of mathematics. Mathematics is an important foundation of philosophical thinking. Its position in scientific culture also makes it an important foundation of philosophical thinking. Many important debates in the field of philosophy in history often involve understanding some basic problems in mathematics. Thinking about these problems will help us to correctly understand the related debates in mathematics and philosophy. (A) Mathematics-rooted in the external performance of practical mathematics, more or less related to human intellectual activities. Therefore, in the relationship between mathematics and practice, mathematics has always been advocated as "the free creation of human spirit", denying that mathematics comes from practice. In fact, all the development of mathematics comes down to practical needs to varying degrees. It can be seen from Oracle Bone Inscriptions of Yin Dynasty in China that our ancestors had already used the decimal counting method at that time. In order to meet the needs of agriculture, they matched "ten branches" and "twelve branches" into sixty jiazi to record the year, month and day. Thousands of years of history show that this calendar calculation method is effective. Similarly, due to the calculation of business and debt, the ancient Babylonians had multiplication tables and reciprocal tables, and accumulated a lot of materials belonging to the category of elementary algebra. In Egypt, due to the need to re-measure land after the Nile flood, a lot of geometric knowledge of calculating area has been accumulated. Later, with the development of social production, especially the astronomical survey met the needs of agricultural cultivation and navigation, elementary mathematics gradually formed, including most of the mathematical knowledge we learned in middle schools today. Later, the industrial revolution triggered by the invention of steam engine and other machinery required a more detailed study of motion, especially variable speed motion, and a large number of mechanical problems appeared, which prompted calculus to appear after a long period of brewing. Since the 20th century, with the rapid development of modern science and technology, mathematics has entered an unprecedented period of prosperity. During this period, many new branches of mathematics appeared: computational mathematics, information theory, cybernetics, fractal geometry and so on. In short, the need of practice is the most fundamental driving force for the development of mathematics. The abstraction of mathematics is often misunderstood by people. Some people think that the axioms, postulates and theorems of mathematics are only the products of mathematicians' thinking. Mathematicians work on a piece of paper and a pen and have nothing to do with reality. In fact, even as far as Euclid's geometry, the earliest axiomatic system, is concerned, the geometric intuition of practical things and the phenomena developed by people in practice, although they do not conform to various axiomatic systems of mathematicians, still contain the core of mathematical theory. When a mathematician aims at the establishment of a geometric axiom system, his mind is bound to be linked with geometric drawing and intuitive phenomena. A person, even a talented mathematician, can get scientific results in the study of mathematics. In addition to strict mathematical thinking training, he will be consciously or unconsciously guided by practice in the process of mathematical theory research, such as asking questions, choosing methods, prompting conclusions and so on. It can be said that without practice, mathematics will become passive water and a tree without roots. In fact, even in Euclidean geometry, the earliest axiomatic system, the geometric intuition of practical things and the phenomena found in practice, although not in line with the procedures of mathematician's axiomatic system, still contain the core of mathematical theory. When a mathematician aims at the establishment of geometric axiom system, his mind is bound to be linked with geometric drawing and intuitive phenomena. A person, even a talented mathematician, can get scientific results in the study of mathematics. In addition to strict mathematical thinking training, I will be consciously or unconsciously guided by practice in the process of mathematical theory research in many aspects, such as asking questions, choosing methods and prompting conclusions. It can be said that without practice, mathematics will become passive water and a tree without roots. However, due to the characteristics of mathematical rational thinking, it will not be satisfied with only studying the realistic quantitative relations and spatial forms, but also trying to explore all possible quantitative relations and spatial forms. In ancient Greece, mathematicians surpassed the method of measuring line segments within the limited calibration accuracy and realized the existence of incommensurable measuring line segments, that is, the existence of irrational numbers. This is actually one of the most difficult concepts in mathematics-continuity and infinity. It was not until two thousand years later that the same problem led to the in-depth study of limit theory and greatly promoted the development of mathematics. Imagine what we would face without the concept of real numbers today. At this time, people can't measure the diagonal length of a square, and they can't solve a quadratic equation: as for limit theory and calculus, it's even more impossible. Even if people can apply calculus like Newton, they will feel at a loss when judging the truth of the conclusion. How far can technology go in this case? For example, when Euclidean geometry was produced, people doubted the independence of one of the postulates. In the first half of19th century, mathematicians changed this postulate and got another possible geometry-non-Euclidean geometry. The founder of this geometry showed great courage, because the conclusion drawn by this geometry is very "absurd" from the perspective of "common sense". For example, "the area of a triangle will not exceed a positive number." There seems to be no such geometric position in the real world. But nearly a hundred years later, in the theory of relativity discovered by physicist Einstein, non-Euclidean geometry is the most suitable geometry. For another example, in the 1930s, Godel got the result that mathematical conclusions were uncertain, and some of the concepts were very abstract, but in recent decades, they have found applications in the analysis of algorithmic languages. In fact, many applications of mathematics in some fields or some problems, once the practice promotes mathematics, mathematics itself will inevitably gain a kind of power, which may exceed the boundaries of direct application. This development of mathematics will eventually return to practice. In a word, we should vigorously advocate the study of mathematical topics directly related to the current practical application, especially in the real economic construction. But we should also establish an organic connection between pure science and applied science, and establish a balance between abstract personality and colorful personality, so as to promote the coordinated development of the whole science. (2) Mathematics-full of dialectics. Because of the rigor of mathematics, few people doubt the correctness of mathematical conclusions. On the contrary, mathematical conclusions often become the model of truth. For example, there is no doubt that people often use "as sure as one plus one equals two" to express their conclusions. In the teaching of our primary and secondary schools, mathematics is only allowed to imitate, practice and recite. Is mathematics really an eternal absolute truth? In fact, the truth of mathematical conclusions is relative. Even a simple formula like 1+ 1=2 has its own shortcomings. For example, in Boolean algebra, 1+ 1=0! Boolean algebra is widely used in electronic circuits. Euclidean geometry is always correct in our daily life, while non-Euclidean geometry is suitable for studying some problems of celestial bodies or the motion of fast particles. Mathematics is actually very diverse, and its research scope is constantly expanding with the emergence of new problems. Like all sciences, if mathematicians stick to the ideas, methods and conclusions of their predecessors, mathematical science will not progress. It is wrong to regard the rigor and axiomatic system of mathematics as a kind of "dogma", what's more, scholars in feudal times said to Confucius that "truth" has been included in what saints said, and future generations can only interpret it. The history of the development of mathematics can prove that it is the innovative spirit of mathematicians, especially young mathematicians, who dare to challenge the old-fashioned ideas, which makes the face of mathematics constantly updated, and mathematics has grown into such a vibrant and energetic subject today. The axiomatic system of mathematics has never been an unquestionable and unchangeable "absolute truth". Euclid's geometric system is the earliest mathematical axiom system, but from the beginning, some people suspected that the fifth postulate was not independent, that is, it could be deduced from other parts of the axiom system. People have been looking for answers for more than two thousand years, and finally non-Euclidean geometry was discovered in19th century. Although people have been bound by Euclidean geometry for a long time, they finally accepted different geometric axioms. If some mathematicians in history were more innovative in challenging the old system, the axiomatic system of non-Euclidean geometry might have appeared hundreds of years ago, which reflected the requirement of inherent logic rigor. In a subject field, when the relevant knowledge is accumulated to a certain extent, the theory will need a bunch of seemingly scattered achievements to be expressed in some systematic form. This requires re-understanding, re-examining and rethinking the existing facts, creating new concepts and methods, and making the theory contain the most common and newly discovered laws as much as possible. This is really a hard process of theoretical innovation. The same is true of mathematical axiomatization, which means that mathematical theory has developed to a mature stage, but it is not the end of understanding once and for all. Existing knowledge may be replaced by deeper understanding in the future, and existing axioms may be replaced by a more general axiom system containing more facts in the future. Mathematics is developed in the process of constant renewal. There is a view that applied mathematics is to apply familiar mathematical conclusions to practical problems, and teaching in primary and secondary schools is to teach students these eternal dogmas. In fact, the application of mathematics is extremely challenging. On the one hand, we need to deeply understand the actual problem itself, on the other hand, we need to master the true meaning of relevant mathematical knowledge, and more importantly, we need to creatively combine the two. As far as the content of mathematics is concerned, mathematics is full of dialectics. In the development period of elementary mathematics, metaphysics is dominant. In the eyes of mathematicians or other scientists at that time, the world was made up of rigid things. Accordingly, the object of mathematical research at that time was constant, that is, the constant quantity. Cartesian variable is a turning point in mathematics. He combined geometry and algebra, two completely different fields in elementary mathematics, and established the framework of analytic geometry, which has the characteristics of expressing movement and change, so dialectics entered mathematics. Calculus, which came into being shortly thereafter, abandoned the view that the conclusion of elementary mathematics is eternal truth, often made opposite judgments and put forward some propositions that elementary mathematics representatives could not understand at all. Mathematics has come to such a field that even simple relations have taken a completely dialectical form, forcing mathematicians to become dialectical mathematicians unconsciously and involuntarily. The objects of mathematical research are full of contradictory opposites: curves and straight lines, infinite and finite, differential and integral, accidental and inevitable, infinite and infinitesimal, polynomial and infinite series. Because of this, classical Marxist writers often mention mathematics in their discourses on dialectics. If you learn a little mathematics, it will definitely help you understand dialectics.

7. Test math scores.

Senior high school entrance examination (Jiangsu):

Chinese, math 150, English 150, physics 130, chemistry 100, history 100, politics 50: sports 50, 40.

College entrance examination:

Chinese 150 Mathematics 150 English 150 Literature Comprehensive (Science Comprehensive) 300 with a total score of 750.

It can be seen that mathematics plays an important role in both life and study.

1. References:

Encyclopedia entry "mathematics"

Ophthalmology b

2. Math scores are included in the total score of the cultural examination.

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3. Baidu Encyclopedia "Mathematics and Culture" entry