Handbook of Mathematics: Why do we study mathematics?

Gu Sen: From childhood to college, we always use ready-made knowledge and theorems to solve problems. Many people still don't know the formula or the reason which scientists put forward after graduation. But in fact, the story behind it is even more exciting. Including many mathematical conclusions, in fact, at first, people's guess may be wrong, and even get a completely opposite conclusion, and then gradually approach this truth. These are not in the textbook. There are many books that are purely about stories, and The Silent Universe tells the origin of many mathematical formulas and their influence on human beings, which is very powerful.

I think the physical formula has a greater impact on the significance of human development. In fact, mathematical formulas are not the most meaningful things in mathematics. Most of them are theorems, not what is equal to what.

Li Miao: I think even liberal arts students need to study across borders. I strongly support the reform of the college entrance examination, regardless of arts and sciences. A thousand years ago, there was no division. Under the guidance of human science, we must go back to the past after the division of labor is extremely detailed and science develops to a certain extent. It is to cross the border. If you don't cross the border, your employment opportunities and all aspects will be greatly restricted in the future. So I am very much looking forward to seeing in my lifetime that the college entrance examination will no longer be divided into subjects.

Gu Sen: If it's for the exam, will we still study if the math exam is cancelled? I think mathematics should be divided into two levels. One is to add, subtract, multiply and divide within 1 in primary school, and learn to settle accounts. Further solving equations, geometric conclusions, etc., may not be used in life. It is enough to learn mathematics in the second day of junior high school, and then go up, perhaps purely out of interest. Mathematical physics is meaningful only when you want to do something completely new in a certain professional field and benefit mankind.

A Complete Collection of Mathematical Manuscripts: Behind Great Mathematical Formulas

On May 15th, 1971, Nicaragua issued a set of ten sheets entitled? Ten mathematical formulas to change the face of the world? Stamps, selected by some famous mathematicians, are commended by ten formulas with great influence on world development. These ten formulas not only benefit mankind, but also have typical mathematical beauty, namely, simplicity, harmony and singularity.

(1) Basic rules of finger counting

Stamps? 1+1=2? It is the first stamp in this set, which is the basic formula of human understanding of quantity at the beginning. The ancestors of mankind started with this formula, piling stones, counting shells, branches and bamboo pieces, then scoring and counting knots, and then creating words, numbers and counting appliances such as abacus, calculation and calculator. Everything starts with the basic law of finger counting, because people have ten fingers, and they are assisted in calculation. Undoubtedly, it is this fact that naturally gave birth to the decimal system that we are familiar with now. The birth of notation and decimal system is a leap in the history of civilization.

(2) Pythagorean Theorem (Pythagorean Theorem)

If the right sides of a right triangle are A and B and the hypotenuse is C, then A2+B2=C2, which is the most famous Pythagorean Theorem in Euclidean geometry. It is widely used in mathematics and human practice. Pythagoras, a famous philosopher and mathematician in ancient Greece, gave the first proof of this theorem abroad, so it is generally called? Pythagoras theorem? .

China knew about it in Shang Gao's time? Hook three strands, four strings and five? Relationship, much earlier than Pythagoras, however, China's proof of Pythagorean theorem is a relatively late thing, until the Three Kingdoms period, Zhao Shuang gave its first proof by area cut-and-fill method. One of the great influences of Pythagorean Theorem is the discovery of irrational numbers. The diagonal length of a square with a side length of 1 is , which cannot be expressed by integers or the ratio of integers, that is, fractions. This discovery denies the Pythagorean school? Everything is counted? At that time, people thought that integers and fractions were easy to understand and called them rational numbers, while the newly discovered number was not easy to understand but existed, so it was named? Irrational number? .

(III) Archimedes lever principle

The mathematical formula F1X1=F2X2 commended by the third stamp, where F is the acting force, X is the arm of force, and FX is the moment. In principle, as long as the power arm is long enough and the resistance arm is short enough, a sufficiently heavy object can be pried with a small enough force. To this end, Archimedes said an old saying:? Give me a fulcrum, and I can move the earth? . Hehe, look how confident physicists are! ! ! In addition to the lever principle, Archimedes also discovered the famous law of buoyancy and a large number of geometric theorems, and he was also one of the pioneers of calculus. Be honored by later mathematicians? God of mathematics? Among the three most important mathematicians in human history, Archimedes is the first, and the other two are Newton and Gauss respectively.

(4) the formula of the relationship between Napier index and logarithm

The logarithmic formula is the Napier formula, where e=2.71828? . The inventor of logarithm is Baron Napier, an amateur mathematician in Scotland. Since the age of 44, after 2 years of studying the calculation technology of large numbers, he finally invented logarithm independently. In 1614, he published the famous book "The Wonderful Law of Logarithm", and the amazing invention of logarithmic table quickly spread all over the European continent. Galileo made a magnificent speech: Give me time, space and logarithm, and I can create a universe. ? Logarithmic tables have been widely used by mathematicians, accountants, navigators and scientists for centuries. Logarithm and exponent have become the essence of mathematics, and they are the contents that every middle school student must learn.

(5) Newton's law of gravity

The fifth stamp immediately reminds people of the story of Newton and Apple, which has long been a household name. In that magical holiday, an apple accidentally fell from the tree, which was a turning point in the history of human thought. It opened the mind of the man sitting in the garden, and finally Newton discovered the epoch-making law of gravity for human beings.

where g is the gravitational constant, m1 and m2 respectively represent the mass of two objects, and r is the distance between two objects.

(6) Maxwell's electromagnetic equations

The sixth formula is Maxwell's electromagnetic equations, which determines the general relationship among charge, current, electric field and magnetic field and is the basic equation of electromagnetism. Maxwell's equations show that the eddy electric field can be excited as long as there is a changing magnetic field somewhere in space, and the changing electric field can excite the eddy magnetic field, and the alternating electric field and magnetic field excite each other to form a continuous electromagnetic oscillation, that is, electromagnetic waves. This formula can prove that the speed of electromagnetic wave propagation in vacuum is equal to the speed of light propagation in vacuum, which is not accidental coincidence, but because light is electromagnetic wave with a certain wavelength, which is Maxwell's electromagnetic theory of light. Maxwell is a great physicist who integrates electromagnetism after Faraday. The theory of electromagnetism has laid the foundation of modern electric power industry, electronic industry and radio industry. In 1871, he was appointed as the professor of experimental physics at Cambridge University, and was responsible for the establishment of the first physics laboratory in the university? Cavendish laboratory.

(7) Einstein's mass-energy relation

E=mc2

, where c is the speed of light, m is the mass and e is the energy. This is the most famous mass-energy relationship later. This is the theoretical basis for making an atomic bomb. The person who put forward this formula in 195 was Einstein, a clerk of Bern Patent Office, who was only 26 years old. In 1915, the general theory of relativity was established, and the relationship between space, time and matter was determined. The mass-energy conversion formula and the theory of relativity have great influence. Today, nuclear energy is widely used in agriculture and military affairs, while black holes, time travel and space bending are all derived from the theory of relativity. Einstein studied the violin at the age of 6, and accompanied him all his life. Art improved his aesthetic ability, and he also pursued the beauty of mathematics (simplicity and symmetry) in physics all his life.

(8) De Broglie formula

The formula commended by the eighth stamp is the De Broglie formula for expressing wave-particle duality proposed by De Broglie in 1924. =h/mv,

where? Is the wavelength of the matter wave associated with the particle, H is Planck constant, and mv is the momentum of the particle. Before de Broglie, people's understanding of nature was limited to two basic material types: physical objects and fields. De Broglie originally studied history, but was influenced by the mathematician Poincare and changed to science. In 1924, he put forward the concept of "matter wave" in his doctoral thesis, which caused a sensation all over the world. He believed that any object and particle had both properties of wave and particle, and used Einstein's theory of relativity to derive the formula of matter wave wavelength. His view was later confirmed by Davidson's experiment. The concept of matter wave also provides an important theoretical basis for the development of wave mechanics.

(9) Boltzmann formula

In 1854, the German scientist Clausius first introduced the concept of entropy, which is a quantity to express the disorder degree of a closed system. Is entropy Greek? Change? Meaning of. This quantity will not change in the reversible process, but will become larger in the irreversible process. Just like a lazy man's room, if no one cleans it for him, the room will only go down in disorder and never become neat. Biology can't live without it? Law of entropy increase? Living things need to absorb negative entropy from outside to offset the increase of entropy. In 1877, Boltzmann expressed the disorder of the system with the following relation: S=kLnW, where k is Boltzmann constant and s is the entropy value of the macro system, which is a measure of the degree of molecular movement or disorder. W is the number of possible microscopic states. The greater the w, the more chaotic the system is. From this, we can see the microscopic significance of entropy: entropy is a measure of the disorder of molecular thermal motion in the system. Because of his novel viewpoint, it was not accepted by many famous scholars at first, and Boltzmann paid a huge price for it, which became an important reason for his personal tragedy (suicide). This formula S=kLnW is engraved on Boltzmann's tombstone in recognition of his great originality.

(1) Tsiolkovsky formula

the Goddess Chang'e flying to the moon, thousands of families have soared, and human beings have been longing for space for a long time, and they have made unremitting efforts to this end. The key to conquer space is rocket technology.

When it comes to modern rockets, it is necessary to mention tsiolkovsky, a world-recognized pioneer of space theory, the former Soviet Union. It was he who proposed the possibility of using rockets for interstellar navigation and launching satellites. The relationship between rocket structure characteristics and flight speed is established, that is, the famous Tsiolkovsky formula. Where V is the speed increment of the rocket, Ve is the speed of the jet relative to the rocket, and m and mi represent the mass of the rocket when the engine is turned on and off, respectively. It has become the key to human conquest of space.

In 1957, the Soviet Union launched the first artificial satellite, which opened the prelude to the space age. In 1961, it sent the first astronaut, Gakkarin, and won the first battle of the space race. In 1969, the United States sent Armstrong to the moon. Tsiolkovsky focused on China's ancient rocket technology, and asked people to translate military works in the late Ming and early Qing dynasties for reference, especially interested in Wu Beizhi. At that time, China had nearly 3 kinds of military rockets. God machine fire dragon arrow? Or? Fire dragon out of the water? Weapons like this fascinated him, and he had more dreams and inspirations, and soon wrote a book "Dreams of the Earth and the Sky". He has a very incisive famous saying: The earth is the cradle of human beings, but people cannot live in the cradle forever. ?

A Complete Collection of Mathematical Manuscripts: Mathematical Famous Words

1. Mathematics is the queen of science, while number theory is the queen of mathematics. ? Gauss

2. Can a country's scientific level be measured by the mathematics it consumes? Rao

3. Number theory is the oldest branch of human knowledge. However, some of his deepest secrets are closely related to his most ordinary truth. ? Smith

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

5. Sometimes, you can't get the simplest and most wonderful proof at first, but it is this kind of proof that can go deep into the wonderful connection of higher arithmetic truth. This is the motivation for us to continue our research, and it can make us find something most. ? Gauss

6. A science can only be truly perfect when it successfully applies mathematics. ? Marx

7. I am determined to let go of that only abstract geometry. That is to say, no longer to think about those problems that are only used to practice ideas. I did this to study another kind of geometry, that is, Cartesian geometry, which aims at explaining natural phenomena. 8. A mathematician who has no talent as a poet will never become a complete mathematician. 9. Pure mathematics, at its modern development stage, can be said to be the most original creation of the human spirit. ? Whitehead

1. We can expect that with the development of education and entertainment, more people will enjoy music and painting. However, the number of people who can really appreciate mathematics is very small. ? Bells

11. "Problems are the heart of mathematics. ? PRHalmos

12. This is a reliable rule. When the author of a mathematical or philosophical work writes in vague and abstruse words, he is talking nonsense. ? A? N? Whitehead

13. As long as a branch of science can ask too many questions, it is full of vitality, and the lack of questions indicates the termination or decline of independent development. ? Hilbert

14. Pure mathematics, in its modern development stage, can be said to be the most original creation of human spirit. ? Huaidehai

15. When the number is invisible, it is less intuitive, and when the shape is few, it is difficult to be nuanced. Since the number and the shape are interdependent, how can they be divided into two sides? ? Hua Luogeng

16. A peculiar beauty rules the kingdom of mathematics. This beauty is not as similar as the beauty of art and the beauty of nature, but it deeply infects people's hearts and arouses people's appreciation of her, which is very similar to the beauty of art. ? Cuomo

17. Mathematics? Barrow

18. Although we are not allowed to see through the secrets of the nature, so as to know the real reason of the phenomenon, it may still happen that the necessary fictional assumptions are enough to explain many phenomena. ? The origin of Euler

19. The problem is the heart of mathematics. ? PRHalmos

2. No problem can touch people's emotions as deeply as infinity, and few other concepts can stimulate reason to produce fruitful thoughts as infinity, but no other concepts need to be clarified as infinity. ? Hilbert

21. What is the work of a master? It's extraordinary! ? Galois

22. We appreciate mathematics, and we need it. ? Chen Shengshen

23. Mathematics is a deductive knowledge. From a set of postulates, we can draw a conclusion through logical reasoning. ? Chen Shengshen

24. Mathematicians are actually fascinated. Without fascination, there is no Novalis of mathematics

25. Mathematics is incomparable forever.