golden section

Everyone should be familiar with the "golden section"!

Since the Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, modern mathematicians have come to the conclusion that Pythagoras school had touched and even mastered the golden section at that time. In the 4th century BC, eudoxus, an ancient Greek mathematician, first studied this problem systematically and established the theory of proportion.

When Euclid wrote The Elements of Geometry around 300 BC, he absorbed eudoxus's research results and further systematically discussed the golden section, which became the earliest treatise on the golden section. After the Middle Ages, the golden section was cloaked in mystery. Several Italians, pacioli, called the ratio between China and the destination sacred and wrote books on it. German astronomer Kepler called the golden section sacred. It was not until the19th century that the name golden section gradually became popular. The golden section number has many interesting properties and is widely used by human beings. The most famous example is the golden section method or 0.6 18 method in optimization, which was first proposed by American mathematician Kiefer in 1953 and popularized in China in 1970s.

Perhaps, we have learned a lot about the performance of 0.6 18 in science and art, but have you ever heard that 0.6 18 has an indissoluble bond with the fierce and cruel battlefield of gunfire and bloodshed, and also shows its great and mysterious power in the military? Napoleon the Great, a lean man, never thought that his fate would be closely linked with 0. 18. June, 18 12, is the coolest and pleasant summer in Moscow. After the battle of Borokino, which failed to destroy the Russian army, Napoleon led the army into Moscow at this time. At this time, he is full of ambition and arrogance. He didn't realize that genius and luck were disappearing from him at this time, and the peak and turning point of his career came at the same time. Later, the French army withdrew from Moscow in frustration in the heavy snow and howling cold wind. Three months of triumph, two months of climax and decline, from the time axis, when the French emperor overlooked Moscow through the flame, his foot just stepped on the golden section.

The Parthenon in ancient Greece is a world-famous perfect building with an aspect ratio of 0.6 18. Architects found that the palace designed according to this ratio is more magnificent and beautiful; If you design a villa, it will be more comfortable and beautiful. Even doors and windows designed as golden rectangles will be more harmonious and pleasing to the eye.

Interestingly, this number can be seen everywhere in nature and people's lives: the navel is the golden section of the whole human body, and the knee is the golden section from the navel to the heel. The aspect ratio of most doors and windows is also 0.618. On some plants, the included angle between two adjacent petioles is 137 degrees 28', which is exactly the included angle between two radii that divide the circumference into 1: 0.6 18. According to research, this angle has the best effect on ventilation and lighting of the factory building. The golden section is closely related to people. The latitude range of the earth's surface is 0-90 degrees. If divided into the golden section, 34.38-55.62 is the golden zone of the earth. No matter from the aspects of average temperature, annual sunshine hours, annual precipitation and relative humidity, it is the most suitable area for human life. Coincidentally, this region covers almost all the developed countries in the world.

Observe life more, and you will find the wonderful mathematics in life!

figure

China has an idiom-"as the name implies". Many things can be as the name implies, but there are exceptions. Such as Arabic numerals. When many people hear Arabic numerals, they think they were invented by Arabs. But it turns out that this is not the case. Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9. 0 is a common number in the world. This figure was not created by Arabs, but it can't erase the credit of Arabs. In fact, Arabic numerals originated from Indians and were gradually created by their ancestors in production practice.

In 3000 BC, the number of residents in the Indus Valley was advanced, and the decimal system was adopted. By the Vedic era (65438 BC+0400 BC-543 BC), Aryans had realized the role of numbers in production activities and daily life, and created some simple and incomplete numbers. In the 3rd century BC, a complete set of numbers appeared in India, but there were different writing styles in different places, among which Brahmanism was the typical one. Its uniqueness lies in that each number has a special symbol from 1 ~ 9, from which modern numbers are born. At that time, "0" had not appeared. It was not until the Gupta era (300-500 years) that there was a "0", which was called "Shunya", expressing a black dot "●" and later evolved into "0". This produces a complete set of figures. This is the great contribution of the ancient Indian people to world culture.

Indian figures first spread to Sri Lanka, Myanmar, Cambodia and other countries. In the 7th and 8th centuries, with the rise of the Arab Empire across Asia, Africa and Europe, Arabs eagerly absorbed the advanced cultures of ancient Greece, Rome, India and other countries and translated a large number of their scientific works. In 77 1 year, Indian astronomer and traveler Maoka visited Baghdad, the capital of the Abbasid Dynasty of the Arab Empire (750- 1258), and presented an Indian astronomical work Sidan Tower to the then caliph Mansour (757-775), who translated it into Arabic and named it Sindh. There are many numbers in this book, so it is called "Indian Numbers", which means "from India".

Arabian mathematicians Hua Lazimi (about 780-850) and Haibosh first accepted Indian numerals and used them in astronomical tables. They gave up their 28 letters, revised and perfected them in practice, and introduced them to the west without reservation. At the beginning of the 9th century, Hua Lazimi published "India Counting Algorithm", and expounded Indian numbers and their application methods.

Indian numerals replaced the long and clumsy Roman numerals, which spread in Europe and were opposed by some Christians, but proved to be better than Roman numerals in practice. 1202 The Calculation Book published by Leonardo in Italy marked the beginning of the use of Indian numerals in Europe. Chapter *** 15 of the book says: "The nine numbers in India are' 9, 8, 7, 6, 5, 4, 3, 2, 1', and any number can be represented by these nine numbers and the symbol' 0' called sifr (zero) by Arabs."

/kloc-In the 4th century, printing in China spread to Europe, which accelerated the popularization and application of Indian numerals in Europe and was gradually adopted by Europeans.

Westerners accepted the Indian numerals sent by Arabs, but forgot their founders and called them Arabic numerals.

95 Respondent: He Qian Intern Level 1 2009-8-10/kloc-0: 6: 42

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Mathematics thesis I

About "0"

0, can be said to be the earliest human contact number. Our ancestors only knew nothing and existence at first, and none of them was 0, so 0 isn't it? I remember the primary school teacher once said, "Any number minus itself is equal to 0, and 0 means there is no number." This statement is obviously incorrect. As we all know, 0 degrees Celsius on the thermometer indicates the freezing point of water (that is, the temperature of ice-water mixture at standard atmospheric pressure), where 0 is the distinguishing point between solid and liquid water. Moreover, in Chinese characters, 0 means more as zero, such as: 1) fragmentary; A small part. 2) The quantity is not enough for a certain unit ... At this point, we know that "no quantity is 0, but 0 not only means no quantity, but also means the difference between solid and liquid water, and so on."

"Any number divided by 0 is meaningless." This is a "conclusion" about 0 that teachers from primary school to middle school are still talking about. At that time, division (primary school) was to divide a copy into several parts and figure out how many there were in each part. A whole cannot be divided into 0 parts, which is "meaningless". Later, I learned that 0 in a/0 can represent a variable with zero as the limit (the absolute value of a variable is always smaller than an arbitrarily small positive number in the process of change) and should be equal to infinity (the absolute value of a variable is always larger than an arbitrarily large positive number in the process of change). From this, another theorem about 0 is obtained: "A variable whose limit is zero is called infinitesimal".

"Room 203 105 in 2003", although all of them are zeros, they are roughly similar in appearance; They have different meanings. 0 indicator vacancy of 105 and 2003 cannot be deleted. 0 in Room 203 separates "Building (2)" from "House Number". (3) "(that is, Room 8 on the second floor) can be deleted. 0 also means that ...

Einstein once said: "I always think it is absurd to explore the meaning and purpose of a person or all living things." I want to study all the numbers of "existence", so I'd better know the number of "non-existence" first, so as not to become what Einstein called "absurd". As a middle school student, my ability is limited after all, and my understanding of 0 is not thorough enough. In the future, I hope (including action) to find "my new continent" in the "ocean of knowledge".

Mathematics paper 2

Mathematicization of various sciences

What exactly is mathematics? We say that mathematics is a science that studies the relationship between spatial form and quantity in the real world. It is widely used in modern life and production, and is an essential basic tool for studying and studying modern science and technology.

Like other sciences, mathematics has its past, present and future. We know its past in order to understand its present and future. The development of modern mathematics is extremely rapid. In recent 30 years, the new mathematical theory has surpassed the sum of 18 and 19 th century theories. It is estimated that it will take less than 10 years for each "doubling" of future mathematical achievements.

An obvious trend in the development of modern mathematics is that all sciences are going through the process of mathematization.

For example, physics has long been regarded as inseparable from mathematics. In colleges and universities, it is also a well-known fact that students of mathematics department should study general physics and students of physics department should study advanced mathematics.

Another example is chemistry. We should use mathematics to quantitatively study chemical reactions. We should take the concentration and temperature of the substances involved in the reaction as variables, express their changing laws with equations, and study the chemical reaction through the "stable solution" of the equations. Not only basic mathematics should be applied here, but also "frontier" and "developing" mathematics should be applied.

For example, biology should study the periodic movement of heartbeat, blood circulation and pulse. This movement can be expressed by an equation. By finding the "periodic solution" of the equation and studying the appearance and maintenance of this solution, we can grasp the above biological phenomena. This shows that biology has developed from qualitative research to quantitative research in recent years, and it also needs to apply "developing" mathematics. This has made great achievements in biology.

When it comes to demography, it is not enough just to add, subtract, multiply and divide. When we talk about population growth, we often say what the birth rate is and what the death rate is. So the birth rate minus the death rate is the annual population growth rate? No, in fact, people are constantly born, and the number of births is related to the original base. So is death. This situation is called "dynamic" in modern mathematics. It can't be simply treated by addition, subtraction, multiplication and division, but described by complex "differential equations". Study such problems, equations, data, function curves, computers, etc. Both are indispensable. Finally, it can be clear how each family can have only one child, how to have only two children, and so on.

As for water conservancy, we should consider the storm at sea, water pollution and port design. We also use equations to describe these problems, and then input the data into the computer to find out their solutions, and then compare them with the actual observation results to serve the actual situation. Very advanced mathematics is needed here.

When it comes to exams, students often think that exams are used to check students' learning quality. In fact, the examination methods (oral examination, written examination, etc. ) and the quality of the test paper itself is not the same. Modern educational statistics and educational metrology test the examination quality through quantitative indicators such as validity, difficulty, discrimination and reliability. Only qualified exams can effectively test students' learning quality.

As for literature, art and sports, mathematics is essential. We can see from CCTV's literary and art grand prix program that when an actor is graded, it is often "to remove a highest score" and then "to remove a lowest score". Then, the average score of the remaining scores is calculated as the actor's score. Statistically speaking, "the highest score" and "the lowest score" have the lowest credibility, so they are removed.

Mr. Guan, a famous mathematician in China, said: "There are various inventions in mathematics, and I think there are at least three: one is to solve classic problems, which is a great job; First, put forward new concepts, new methods and new theories. In fact, it is this kind of person who has played a greater role in history and is famous in history; Another is to apply the original theory to a brand-new field, which is a great invention from the perspective of application. " This is the third invention. "There are a hundred flowers here, and the prospects for the development of mathematics and other sciences to comprehensive science are infinitely bright."

As Mr. Hua said in May 1959, mathematics has developed by leaps and bounds in the past 100 years. It is no exaggeration to summarize the wide application of mathematics with "the vastness of the universe, the smallness of particles, the speed of rockets, the cleverness of chemical industry, the change of the earth, the mystery of biology, the complexity of daily life, etc." The greater the scope of applied mathematics, all scientific research can solve related problems with mathematics in principle. It can be asserted that there are only departments that can't apply mathematics now, and they will never find areas where mathematics can't be applied in principle.

Mathematics thesis III

What is mathematics?

What is mathematics? Some people say, "Isn't mathematics the knowledge of numbers?"

That's not true. Because mathematics not only studies "number" but also "shape", triangles and squares, which are familiar to everyone, are also the objects of mathematical research.

Historically, there have been various views on what mathematics is. Some people say that mathematics is relevance; Some people say that mathematics is logic. "Logic is the youth of mathematics, while mathematics is the prime of logic."

So, what is mathematics?

Engels, the great revolutionary tutor, stood at the theoretical height of dialectical materialism, profoundly analyzed the origin and essence of mathematics and made a series of incisive scientific conclusions. Engels pointed out that "mathematics is a quantitative science" and "the object of pure mathematics is the spatial form and quantitative relationship of the real world". According to Engels' point of view, it is more accurate to say: mathematics-a science that studies the quantitative relationship and spatial form of the real world.

Mathematics can be divided into two categories, one is pure mathematics and the other is applied mathematics.

Pure mathematics, also called basic mathematics, specializes in the internal laws of mathematics itself. The knowledge of algebra, geometry, calculus and probability introduced in primary and secondary school textbooks belongs to pure mathematics. A remarkable feature of pure mathematics is to temporarily put aside the specific content and study the quantitative relationship and spatial form of things in pure form. For example, it doesn't matter whether it is the area of trapezoidal rice fields or the area of trapezoidal mechanical parts. What everyone cares about is the quantitative relationship contained in this geometry.

Applied mathematics is a huge system. Some people say that it is the part of all our knowledge that can be expressed in mathematical language. Applied mathematics is limited to explaining natural phenomena and solving practical problems, and it is a bridge between pure mathematics and science and technology. It is often said that now is the information society, and the "information theory" which specializes in information is an important branch of applied mathematics. Mathematics has three most remarkable characteristics.

High abstraction is one of the remarkable characteristics of mathematics. Mathematical theory has a very abstract form, which is formed through a series of stages, so it greatly exceeds the general abstraction in natural science, and not only the concept is abstract, but also the mathematical method itself is abstract. For example, physicists can prove their theories through experiments, while mathematicians can't prove theorems through experiments, but can only use logical reasoning and calculation. Now, even geometry, which used to be regarded as "intuitive" in mathematics, is developing in the abstract direction. According to the axiomatic thought, there is no need to know geometric figures. It doesn't matter whether they are round or square. Even tables, chairs and beer cups can be used instead of dots, lines and noodles. As long as the relationship of combination, order and reduction is satisfied, and it is compatible, independent and complete, a geometry can be formed.

The rigor of the system is another remarkable feature of mathematics. The correctness of mathematical thinking lies in the rigor of logic. As early as more than 2000 years ago, mathematicians started from several basic conclusions and used the method of logical reasoning to organize rich geometric knowledge into a rigorous and systematic theory, just like a beautiful logical chain, with every link connected into a line. Therefore, mathematics has always been regarded as a "model of precise science".

Widely used is also a remarkable feature of mathematics. The size of the universe, the tiny particles, the speed of rockets, the ingenuity of chemical engineering, the change of the earth, the mystery of biology and the complexity of daily life require mathematics everywhere. In the 20th century, with the emergence of a large number of branches of applied mathematics, mathematics has penetrated into almost all scientific departments. Not only physics, chemistry and other disciplines are still enjoying the fruits of mathematics widely, but even biology, linguistics and history, which rarely used mathematics in the past, are combined with mathematics, forming rich marginal disciplines such as biomathematics, mathematical economics, mathematical psychology, mathematical linguistics and mathematical history.

Mathematicization of various sciences is the main trend of the development of modern science.