-Extract mathematical culture from multiple angles-Note 1

Dazhai No.1 Middle School? Gao Yuan Street.

order

Learning mathematics is "study hard+test" and "calculation+logic" Therefore, the current model of top students in middle school mathematics education is that they have a solid foundation, narrow knowledge, can overcome problems without strong creative ability, and have poor ability to solve practical problems by hands and applying mathematics. They are "competitive" but lack "curiosity" about things, so they lack the ability to innovate.

Mathematics is an important part of human culture. Mathematics curriculum should reflect the history, application and development trend of mathematics, as well as the mathematician's ideological system, aesthetic value, innovative spirit and the role of mathematics in the development of human civilization, so as to gradually form a correct view of mathematics among students.

Mathematics is not only a science, but also a culture, that is, "mathematical culture"; Mathematics is not only some knowledge, but also a quality, that is, "mathematical quality". Mathematical culture is an important part of modern humanistic quality.

Professor Qi said: "Without modern mathematics, there would be no seat belt culture; Without modern mathematics, culture is doomed to decline. "

The history of a science is the most precious part of this science, because science can only give us knowledge, while history can give us wisdom.

If we talk about the history of mathematics, we and our students can feel the beauty and profundity of mathematics, understand the past, present and future of mathematics, worry about the stagnation of mathematics and applaud the progress of mathematics.

Feel the ups and downs of mathematics, you can also feel the ordinary and great personality charm of mathematicians, and experience the joys and sorrows of mathematicians engaged in mathematical research, stumbling on the road of mathematics, and the spirit of persistent pursuit of mathematics.

introduce

What is mathematics?

The problem is the core of mathematics. Halmos.

One example is more effective than ten theorems. newton

The unity of philosophy and mathematics: a beautiful dream. Descartes

Mathematicians are fascinated by nature. Without infatuation, there is no math. -Knowles

The great mathematical conjecture through the ages! -Hua

Galileo: Mathematics is the word God used to write the universe.

Einstein: Mathematics is an art. If you make friends with it, you will understand that you can never live without it.

Pythagoras said, "Numbers rule the universe."

Scientific theory (Gauss): "Mathematics, the queen of science; Number theory, the queen of mathematics. "

Judging from the subject structure of mathematics, mathematics is a model.

From the process of mathematics, mathematics is reasoning and calculation.

Judging from the form of expression of mathematics, mathematics is a symbol.

In terms of the guidance of mathematics to people, mathematics is methodology.

From the value of mathematics, mathematics is a tool.

Characteristics of mathematics

Abstraction of content

Rigidity of reasoning and clarity of conclusion

Extensiveness of application

Overview of mathematical culture

In a narrow sense, "culture" only refers to knowledge, which means that educated people say they have knowledge; On the other hand, "culture" in a broad sense is the accumulation of human material wealth and spiritual wealth. (9 pages)

Mathematics is not only a series of digital symbols, but also contains the connotation of humanistic spirit, which embodies the spirit of seeking truth, courage, cooperation and dedication of human beings. These are the sublimation of humanistic spirit, which is a manifestation of culture. (9 pages)

The main characteristics of mathematical culture are:

Thinking (2) Quantification (3) Development (4) Practicality (5) Education (10 Page)

Connotation of mathematical culture:

Rational spirit of mathematical culture

Humanistic spirit of mathematical culture

The embodiment of the application of mathematical culture

The Relative Stability and Continuity of Mathematical Culture

Reflection, Criticism and Perfection of Mathematical Culture

The world of mathematical culture? (Page 10- 12)

The value of mathematical culture;

Mathematics is an accurate thinking tool.

Mathematics is a scientific language.

Mathematics is the art of reasoning.

Mathematics is an important part of human culture (13- 14)

Mathematics Culture in Mathematics Discipline

2. 1? Mathematical problems arising from the golden section

1. Golden numbers are "numbers given by God", "beautiful numbers" and "beautiful passwords". There are still many "secrets" to be revealed in this "beautiful password". (20 pages)

2. The golden section is a transformation, but it is ever-changing, and all roads lead to the same goal. No matter from which angle, it is exquisite and just right; No matter from which aspect, it has profound connotation, meaningful significance and eternal charm. (27 pages)

3. The golden section embodies the unity of science and art, sensibility and rationality, image thinking and logical thinking, and is a masterpiece of human understanding of the world. The golden section is a "beautiful password" given by God. (27 pages)

2.2? Mysterious infinite world

1. Infinity is an eternal mystery, and mathematics is an infinite science. (27 pages)

2. Infinity is both real infinity and latent infinity. Infinity itself is a contradiction, which is not only a process requiring infinite approximation, but also an entity that can be studied. As Professor Xu Lizhi, a famous mathematician in China, said, "Infinity in reality and infinity in potential are just two sides of the same coin." (34 pages)

Hilbert said: "Infinity is not only the greatest friend of mankind, but also the greatest enemy of human peace of mind." (36 pages)

2.3? Appreciation of Pythagorean Theorem

1. The proof of Pythagorean theorem is the beginning of demonstrating mathematics. It is the first theorem in history that connects shapes with numbers, that is, the first theorem that connects geometry with algebra, and it is also the best "language" that mathematicians think to explore alien civilization and communicate with aliens. (47 pages)

2. China's mathematical cultural tradition embodies the pragmatic spirit of attaching importance to application, combining numbers with shapes and focusing on calculation. (47 pages)

2.4? -An endless song.

1. German mathematician Cantor once pointed out: "The accuracy of pi can be used as a symbol to measure the mathematical level of a country." (48 pages)

2. Calculate the value. In addition to the geometric method, analytical method and computer mentioned above, there is also a strange method that does not need complicated calculation-experimental method. (56 pages)

3. Pi is like a maze, which makes people linger; Skin is like a hazy poem, like a melodious regional movement, and like a mountain peak in the clouds, which makes people daydream, intoxicated, forge ahead and climb endlessly! (Page 6 1)

2.5? China remainder theorem

1. "Shu Shu Jiu Zhang" said: "Mathematics is subtle and it is not easy to see it. When I was poor, I was ambitious and felt very dreamy. Fortunately, I learned that I won't hide it. " (65 pages)

2.6? The problem of seven bridges and a sum

1. The so-called "graph theory" is a new discipline that uses intuitive graphics and mathematical methods to study combinatorial relations. (69 pages)

2.7 Three Drawing Problems in Geometry

1. Thick stone walls and solid prison doors have imprisoned Alaxagora's freedom of action, but not his free thoughts. (73 pages)

2.8? The sum of two transcendental irrational numbers

1. In this formula, among the five golden flowers, 0 and 1 come from arithmetic, I from algebra and E from analysis. They are wonderfully blooming at the same time. The two most famous transcendental numbers e go hand in hand, and the real number and imaginary number are dissolved in one furnace. It is well-deserved to call it "the most beautiful formula in mathematics". (86 pages)

2.9? Mobius belt and Klein bottle

1. Mobius tape is made by twisting one end of a rectangular paper tape and then connecting it first. (87 pages)

2. Mobius belt, how simple but extremely profound, has so many wonderful applications in industry and science and technology, and also brings so many novel imaginations to scientists, philosophers, artists and writers. Therefore, we say that Mobius belt is a scientific artistic image and also a science of artistic image. (94 pages)

Chapter three? Mathematical culture in books on the history of mathematics

3. 1? Euclid and Elements of Geometry

1. The Pythagorean school studied under Thales, advocated explaining everything with mathematics, and put forward that "everything has a number". Abstract mathematics from concrete things and establish your own theoretical system. (95 pages)

2. Introduction to the Elements of Geometry: At that time, the mathematical knowledge was systematized by axiomatic methods. The theoretical outline is divided into 13 volumes, including 5 postulates, 5 axioms, 1 19 definitions and 465 propositions, which constitutes the first mathematical axiom system in history. The contents of each volume can be roughly divided into the following categories.

Volume one? Geometric basis?

Volume two? geometric algebra

Volume three? Round?

Volume four? regular polygon

Volume five? Proportional theory?

Volume six? Similar graphics?

Eight or nine volumes? Elementary number theory?

Volume 10? Unusual measurements?

Volumes 1 1, 12 and 13? Solid geometry (pages 97-98)

3. The Elements of Geometry is a major factor in the emergence of modern science. The achievement of great scientific achievements, on the one hand, requires the combination of experience and experiment, on the other hand, requires detailed analysis and deductive reasoning. (98 pages)

Einstein praised the Elements of Geometry and said, "The world witnessed the miracle of a logical system for the first time. This logical system is advanced step by step accurately, and every proposition of it is beyond doubt-I am talking about this amazing victory of Euclid's conjecture and reasoning, which gives human reason the confidence necessary for future achievements. (99 pages)

The Elements of Geometry has the following defects:

This definition is imprecise and strict.

The axiomatic system is incomplete.

The whole book system is incomplete.

The structure of this book is unreasonable.

Some proofs are partial (page 99- 100)

The translation and dissemination of Geometry Elements in China is not only the dissemination of knowledge, but also the dissemination of a scientific method. (Page 103)

Xu Guangqi's achievements in mathematics;

This paper discusses the reasons why China's mathematics fell behind in Ming Dynasty.

The universality of mathematical application is discussed. That is, (1) astronomical calendar; (2) Water conservancy projects; (3) temperament; (4) the art of weapons and military engineering; (5) Accounting and financial management; (6) Various construction projects; (7) mechanical manufacturing; (8) domain measurement; (9) medicine; (10) Make timers such as clock leakage.

Translation of geometric elements. (Page 103)

Liang Qichao called The Elements of Geometry "the immortal work of pure gold and beautiful jade". (Page 104)

In order to further publicize the Elements of Geometry, Xu Guangqi wrote The Elements of Geometry. At the beginning, he said: "study hard and you have reason to do one thing." This book is just right, which can make readers get rid of arrogance and practice care; Learners learn from their own laws and make their own smart thinking. " He worships this book very much, and "thinks" this book has four points: no doubt (doubt), no speculation (guess), no attempt (experiment) and no change (change); There are four impossibilities: unable to get rid of (leave or disappear), unable to refute (refute), unable to reduce (reduce), unable to change the sequence. "(page 104)

Euclid advocated that learning must be gradual and diligent. We don't approve of opportunism and quick success, and we also oppose narrow practical views. (Page 106)

Structural characteristics of geometric elements;

? The first is a closed deductive system.

? The second is content abstraction.

? The third is the axiomatic method mentioned above. (Page 107)

3.2? Liu Hui and Nine Chapters of Arithmetic

1. Nine chapters on arithmetic are very rich. The book is in the form of problem sets, with a total of 246 applied problems related to production and life practice, which are divided into nine chapters according to different contents, which is the origin of the title of Nine Chapters Arithmetic. Each of these questions has questions (topics), answers (answers) and skills (steps and methods of solving problems). Some have one problem and one skill, some have many problems and one skill, and some have many skills. "Shu" is actually an available algorithm, which uses calculation as a tool and is an algorithm for calculating layout. (Page 108)

2. The name and main contents of "Nine Chapters"

? Chapter one? Fang Tian

? Chapter two? corn

? Chapter three? decline

? Chapter four? beautiful springtime

? Chapter five? Commercial workers

? Chapter six? Average loss

? Chapter seven? Excess or shortage

? Chapter eight? equation

? Chapter 9? Pythagoras School (108 page)

3. Liu Hui's "Circumcision": "If you cut it carefully, you will lose less. Cut it, cut it, so that it can't be cut, then it is in harmony with the circle, and nothing is lost. " (Page 1 15)

4. The mathematical thought and cultural significance of Nine Chapters Arithmetic;

(1) open induction system

(2) Algorithm generalization

? (3) Modeling method

? (4) the embodiment of the doctrine of the mean

? (5) Association with Confucian classics (12 1- 122 pages)

3.3? Zhouyi and Binary System

1. I ching includes I ching and I ching. (Page 123)

2. The biggest feature of the Book of Changes is that it has changed the humanistic culture of the Book of Changes: from superstition to rationality, from witchcraft to philosophy. (Page 133)

The Book of Changes shows us a brand-new panorama of the universe. Under the command of "father", "heaven" and "mother", the universe is endless, biochemical and full of infinite vitality and vitality. (Page 133- 134)

4. The cultural value of Zhouyi;

? (1) Yi Xue Chang He

? (2) the cosmic picture

? (3) the origin of mathematics

? (4) Great ideas

? (5) Cultural Seeds (133- 134)

Chapter four? Mathematical culture in mathematical historical materials

4. 1? Paradox and Three Mathematical Crises

1. The usual form of rebellion is: "If you admit that a proposition is correct, you will infer that it is wrong; If you think it is incorrect, you will conclude that it is correct. " Paradox often leads directly to "mathematical crisis". (Page 135)

2. Several famous paradoxes: (1) liar paradox; (2) The paradox of God's omnipotence; (3) The barber paradox; (Page 136)

3. Hebesos paradox and the first crisis: Becquerel paradox and the second mathematical crisis;

Russell paradox and the third mathematical crisis. (137- 14 1 page)

4.2? The Bridge between Geometry and Algebra —— Analytic Geometry

1. Descartes likes this mathematics hall very much, and every proof here is like a bright pearl that people can't put it down. However, Descartes found that people can only pick up these beads, but it is difficult to put on these unique beads. Descartes advocated that all the beautiful things in algebra and geometry should complement each other, so he set out to find a new way to connect algebra and geometry. (Page 146)

4.3? noneuclidean geometry

1. Einstein put forward the theory of relativity and applied Riemann geometry as a mathematical tool, thus unifying the two revolutions in the fields of mathematics and physics. According to the theory of relativity, real space is not evenly distributed, but curved. (Page 158-259)

2. The influence of non-Euclidean geometry is enormous, and it is a revolution in the history of mathematics. It makes mathematicians fundamentally change their understanding of the essence of mathematics and the understanding of mathematics and the material world, and makes people realize that there is an essential difference between mathematical space and physical space. It breaks the belief that mathematical truth is absolute truth, thus making mathematics lose its certainty and truth, but mathematics gains freedom from it, and mathematicians can explore and construct any possible axiomatic system as long as such research is meaningful. (160 page)

4.4? The crystallization of human thinking-calculus

1. Newton is also very excited about returning to school. He wrote a poem entitled "Three Crowns", expressing his willingness to suffer for scientific dedication:

Oh, the worldly crown, I despise it, like the dust under my feet,

It is heavy, and the best is just an emptiness;

But now I'm happy to welcome the crown of thorns,

Although it hurts, the taste is mainly sweet;

I saw the crown of glory in front of me,

It is full of happiness and eternity. (Page 166)

Leibniz commented on Newton in this way: "In all mathematics from the beginning of the world to the time when Newton lived, Newton's work exceeded half." Pope, a famous British poet, described this great scientist like this:

Nature and its laws,

Immersed in chaos,

God said, Newton was born,

Everything becomes clear. (Page 166)

The poet Wordsworth wrote this poem in front of Newton's statue:

Where the statue stands.

It was Newton, with a serious and silent face.

Marble always marks his heart.

Sailing alone in the wonderful ocean of thoughts (167)

Newton himself was very modest. He said: "I don't know who the world thinks of me, but for myself, it's like a child playing by the sea." Sometimes I am very happy to find a relatively smooth pebble or a particularly beautiful shell, and there is an undiscovered sea of truth in front of me. " (Page 167)

With reverence, people carved the following words on his tombstone: "He first explained the movements and images of planets, the orbits of comets and the tides of the sea with almost divine thinking ability. Let ordinary people be happy, because there is an outstanding person among them! " (Page 167)

Cultural significance of calculus;

The function of mathematics itself

The role of other disciplines and engineering technology

Influence on human material civilization

Impact on human culture (174- 175)

Mathematical culture in famous mathematical problems

Horizon: Fermat's Last Theorem

In the history of mathematics, Fermat is known as "the king of amateur mathematicians". (Page 176)

Fermat created the most profound mystery in the history of mathematics! (Page 177)

I have never seen such a wonderful lecture, full of wonderful and unheard-of new ideas, as well as dramatic foreshadowing, full of suspense, until it finally reached its climax. When everyone finally realized that they were only one step away from proving Fermat's last theorem, the air was full of tension. (18 1 page)

When talking about his affection for Fermat's Last Theorem, wiles said, "This is the romance of my childhood, and nothing can replace it ... If you can solve things that are very important to you as an adult, then nothing can be more meaningful." (Page 183)

After completing the proof of Fermat's last theorem, wiles said, "... that special and long exploration has ended, and my heart has returned to peace." (Page 183)

"Fermat's last theorem is like a dazzling gem. It is hidden in the grass in the deep mountain canyon. I was seen by accident. Because of its beauty, it attracted many people to get it, and many people even fell into the abyss. But on the way to conquer it, people discovered rich mineral deposits. This mineral deposit is not Aladdin's treasure house. Everything in it is not necessarily a bright gem, but it can bring a new industrial sector. Without this mineral deposit, this gem can be priceless, but with this mineral deposit, it has become a part of human civilization together with other mineral deposits. " (Page 185)

Goldbach's Conjecture

Goldbach conjecture is "the jewel in the crown of mathematics". (Page 185)

Chen Jingrun trudged on the rugged mathematical mountain road, struggling to keep pace. On the plateau of abstract thinking, he climbed a steep cliff, descended and rose again! Eating frost and drinking snow, the last step is the last step! Panting and sweating, he often felt that he couldn't support it, but he climbed up. It's really hard to use limbs and fingers and claws! Up and down many times, even the iron shoes have been worn out. (190 page)

Writer Xu Chi, with poetic language, praised the pages of the paper, the intoxicating beauty of mathematics! "What a touching page after page! These are the flowers of human thought. This is the orchid in the empty valley, the cuckoo in the cold night, the ginseng in the old forest, the snow lotus on the iceberg, the ganoderma lucidum on the top of the mountain, and the peony with abstract thinking. ..... (192 page)

5.3? Four-color conjecture

1. The British mathematician Seward's spirit of being healthy and vigorous, tireless and tenacious in tackling key problems is worth learning. (Page 196)

2. Up to now, there are still many mathematicians and even amateur mathematicians who are not satisfied with the achievements made by computers and are looking for a logical proof of the four-color problem. (Page 199)

5.4? Hilbert's 23 Mathematical Problems and Their Influence

1. Hilbert pointed out in his speech: "The treasure of mathematical problems is endless. Once a problem is solved, countless new problems will take its place. " He also pointed out: "Every step of real progress in mathematics is closely related to the discovery of more powerful tools and simpler methods, which will help to understand existing theories and throw away old and complicated things. ..... "(204 pages)

5.5? Seven mathematical problems in 2 1 century and their repercussions

1. The seven difficult problems of offering a reward are like "Mount Everest" in the field of mathematics. In the government of Mount Everest, although only a few people finally reached the summit, the survival equipment and skills left by the successful summit will benefit countless latecomers. . Devlin believes that this is the significance of putting forward the seven difficult problems of offering a reward. (205 pages)

2. We firmly believe that the solution of these reward problems will be similar to opening a new world of mathematics that we have never imagined. (209 pages)