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!
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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.
Math is very useful.
Learning mathematics should be used in real life. Mathematics is used by people to solve practical problems. In fact, there will be math problems in life. For example, when you go shopping, you will naturally use addition and subtraction, and you always have to draw pictures when you build a house. There are countless problems like this, and this knowledge comes from life and is finally summed up as mathematical knowledge, which solves more practical problems.
I once saw a report that a professor asked a group of foreign students, "How many times will 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 set hands; 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 and mastering mathematics knowledge in real life.
From then on, I began to consciously connect mathematics with daily life. Once, my mother baked a cake, and two cakes could be put in the pot. I thought, isn't this a math problem? It takes two minutes to bake a cake, one minute in front and one minute in the back. At most, two cakes are put in the pot at the same time. How many minutes does it take to bake three cakes at most? I thought about it and came to the conclusion that it takes 3 minutes: first, put the first cake and the second cake into the pot at the same time, 1 minute later, take out the second cake, put the third cake and turn the first cake over; Bake again 1 min, so that the first cake is ready. Take it out. Then put the reverse side of the second cake on it, and turn the third cake upside down at the same time, so it will be all done in 3 minutes.
I told my mother about this idea, and she said, actually, it won't be so coincidental. There must be an error, but the algorithm is correct. It seems that we must apply what we have learned in order to make mathematics serve our lives better.
Mathematics should be studied in life. Some people say that the knowledge in books has little to do with reality now. This shows that their knowledge transfer ability has not been fully exercised. It is precisely because they can't understand it well and apply it to daily life that many people don't attach importance to mathematics. I hope that students can learn mathematics in their lives and use mathematics in their lives. Mathematics is inseparable from life. If they study thoroughly, they will naturally find that mathematics is actually very useful.
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.
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".
Collect and reprint the solved problems to the mathematics and culture papers in QQ space, with about 3000 words.
200 [Label: Cultural Papers, Mathematical Papers] Linguistic papers can be the history and development of mathematics, the relationship and influence between mathematics and other fields. Anonymous answer: 3 popularity: 1 1 Resolution time: 2008-11-17.
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, which was first published as an axiomatic system, the geometric intuition of practical things and the phenomena found in practice, although not in line with the procedures of the axiomatic system of mathematicians, 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: "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.
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