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The meaning of sx in the Math.sx domain name is (mathematical pinyin abbreviation)
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The essence of mathematics lies in its freedom ——Cantor
The ancient Greek geometer 阿波洛尼乌斯 summarized the theory of conic sections, which was applied to the theory of planetary orbits by the German astronomer Kepler 1,800 years later.
The mathematician 伽罗华 founded group theory in 1831, and it was applied in physics more than a hundred years later.
The matrix theory created in 1860 AD was applied to quantum mechanics sixty years later.
Mathematicians J.H.莱姆伯脱, 高斯, 黎曼, 罗马切夫斯基 and others proposed and developed non-Euclidean geometry. Gauss spent his life exploring the practical applications of non-Euclidean geometry, but he died with regret. One hundred and seventy years after the birth of non-Euclidean geometry, this then useless theory and the theory of tensor analysis developed from it became the core foundation of Einstein's general theory of relativity.
Some people will always say that some theories are useless. However, it turns out that if no one reaches the mountains first, no one dares to look up at the stars, and no one will win the stars and change our real life.
Hardy once said that "beauty is the first touchstone: ugly mathematics cannot last forever". The beauty of mathematics lies in its profound definition, clear logic and perfect result. In basic topology, there is a very nice classification theorem for 2-manifolds - "Any closed surface is either homeomorphic to a sphere, or homeomorphic to a sphere to which a finite number of ring handles are added, or homeomorphic to a dig spheres replaced by Möbius bands without a finite number of discs, and any two of these spheres are anisomorphic to each other"
By the way, this generalizes to n-dimensional spaces like closed surfaces, which we call manifolds. Classification theorems for manifolds of arbitrary dimensions have not been solved so far. The famous Poincaré conjecture is about the homeomorphism of 3-manifolds, but that is another story.
Based on the definition of topological space, homeomorphism and other concepts, mathematicians have carried out logical derivation, thus completing the classification of arbitrary closed surfaces, showing the the profundity and beauty of the light of human reason.
Secondly, Mathematics is a tower built by various top geniuses from ancient times to the present
The charm of mathematics to me is also the personality charm of all kinds of geniuses and diligent and persistent people throughout the ages, mathematics masters who specialize in one field and understand the whole picture of mathematics.
Sometimes when I open the math book at night and look at the legendary names one by one, I seem to be able to see through time and space those figures who were lying at the desk and writing, or frowning and thinking.
It is the mathematical edifice they built that updated human beings' rational understanding of space, measurement, structure, and transformation. When their theories were applied to practice, they changed the overall picture of human life. It also allowed me to express my admiration on the wiki homepage today.
Study mathematics, people who study mathematics always have a pair of quiet eyes, it is because they have seen too many scenery. Opening a math book is like stepping into the door of knowledge
Finally, a sentence ends
If someone doesn't think mathematics is simple, it's because he hasn't realized how complicated life is - Von Neumann
This wiki is a dedicated mathematics wiki, do not write entries that have nothing to do with mathematics, delete them if someone finds them
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Here's how to use this wiki correctly
Edit the main content of the page:http://math.sx/index.php?title= XXX&action=edit
The following are the major mathematical branches of this wiki
- Set Theory
- Mathematical Logic
- Number Theory
- Graph Theory
- category theory
- Elementary Mathematics
- advanced mathematics
This wiki is committed to building the largest mathematical database in China and even in the world
write at the end
Since everyone has read this far, let's take a look at the final story
What is the charm of mathematics?
If you want to understand the charm of mathematics, I recommend a book called "Speechless Universe".
Although there is the word universe in the name, this book is not about astronomical knowledge, but 24 mathematical formulas that change the world. According to author Dana MacKenzie, mathematics is not only a tool widely used in all sciences, but also a common language for expressing knowledge of the universe.
In today's world, no one will deny the importance of mathematics to our work and life. Mathematics can help aunts who buy vegetables to quickly calculate addition, subtraction, multiplication and division, and can also help investment bank managers evaluate option prices; it can teach AlphaGo to learn to play Go, and it can also make Musk's Falcon 9 launch vehicle land vertically. However, if you only see mathematics as a tool to solve problems, then you don't see the way mathematics unfetters our minds; There is a subtle connection between the universe and mathematics. The value of mathematics lies not only in its practical value in solving problems and creating wealth, but also in broadening human horizons; in the long journey of human civilization for thousands of years, mathematics is like a navigator's compass, leading We explore the unknown nature and discover the bigger world. I think this is the way of mathematics that the author wants to tell us through the 24 mathematical formulas that have changed the world from ancient times to the present.
Everything starts with a gamble 70 years ago...
Bet is bet
On a leisurely afternoon in Rio de Janeiro in 1951, the Nobel Prize-winning American theoretical physicist Richard Feynman was eating at a local restaurant he frequented. Professor Feynman, who was in his early thirties, had just participated in the Manhattan Project. In 1945, the first atomic bomb in the United States was successfully tested. This time, Professor Feynman, who was tired of playing with the atomic bomb, came to Brazil for a public tour.
It never occurred to her that this rare vacation time was messed up by a salesman who entered the market. He walked into the restaurant and challenged everyone, claiming that no one could match his abacus when it came to arithmetic. This was of course a salesman's routine, and the waiters booed him to prove that he could do arithmetic with an abacus faster than a physics professor. In this way, one uses a calculator, and the other only relies on pure mathematical skills to do mental calculations. Perhaps the first "strongest brain" PK in history between the abacus salesman and the master of physics began.
The game was completely one-sided at first. When doing addition and subtraction, the abacus salesman reported the answer before Feynman could write the numbers down on paper. The high-spirited salesman proposed to compete with Feynman again. Feynman still lost in this game, but he lost not as badly as the first time. The salesman, a little surprised that he didn't win outright, kept challenging Feynman on harder and harder problems, only to see his edge shrink. As the game became more intense, the salesman had forgotten the performance indicators for this month, and the professor also forgot that he was actually here for dinner. The agreed game was upgraded to a battle of honor between men. In the end, the two decided to use the last problem to decide the winner: calculate the cube root of 1729.03.
The abacus master immediately went into battle, and his fingers were flying on the abacus. Feynman, meanwhile, sat motionless. A few seconds later, Feynman wrote the answer: 12.002. After a few minutes, the salesman reported "12.0", and at this time Feynman's decimal point had added a few more digits. At the end of this betting game, the physicist won a big victory, and the poor salesman was defeated by the waiter's ridicule.
If this true story were put into today, it would definitely make headlines, such as: "Amazing! The ultimate showdown in the man-machine war, the Nobel Prize winner dominates the most powerful brain"; another example: "Ordinary people's brains are only developed 10%, geniuses in physics Scientists reveal human potential"... However, Feynman said in his autobiography: As long as they have a certain amount of mathematical knowledge, ordinary people can also complete "divine operations" such as mental calculation of cube roots. The technique taught by Feynman may seem like magic, but when you unravel the mystery, you will find that it is not mysterious at all on the mathematical level.
Isn't that the cube root of 1729.03? First, you should know that 1728 is 12 cubed. For Feynman, this is common sense, because European and American countries use imperial units, 1 foot = 12 inches, 1 cubic foot = 1728 cubic inches. Then, 1729.03=1728+1.03, since the cube root of 1728 is 12, the cube root of 1729.03 should be a little more than 12. The question is how much? In fact, we don't need to find the exact value, we only need to know the approximate value of 3 digits after the decimal point is enough. If you have studied some basic calculus, you must have thought that there is a famous formula prepared for approximation: Taylor expansion. Expand the function of the cube root of y=x around x=1728, and then bring it into 1.03, and the rest is the four arithmetic operations of addition, subtraction, multiplication and division. The same technique can also be used in the program "The Most Powerful Brain" to find the square root of a large number many times. For example, the 14th root of a 13-digit number looks scary, but it is only used in calculus. Basic knowledge, use the approximate method to estimate the range of the answer. The real difficulty of mental arithmetic is not any advanced mathematical skills, but to memorize the power and root of a large number of numbers, and store as many basic calculation results as possible in the brain and call them at any time.
Advanced mathematics, which has made countless college students sigh "what's the use of learning this", actually exerts its power all the time. However, in today's era of exponentially increasing machine computing power, the mechanical abacus that relies on hands to toggle has long been replaced by semiconductor chips and software algorithms, and no strongest brain can beat a 20-dollar calculator. Does this mean that mathematics is a subject that will eventually be replaced by machines, and it is meaningless for humans to master mathematics?
of course not. Because the mathematical concept behind the formula is much more important than applying the formula and calculating the correct result. It is these concepts that allow human beings to walk all the way from the era of raw hair and blood, and grasp the power to change the world. "Speechless Universe" tells such a story: Behind the 24 mathematical formulas, it is human's efforts to transform their brains for thousands of years in an attempt to understand the universe.
1+1=2, the oldest formula, established our entire mathematical system. Everything from number theory to calculus begins with this kindergarten-level addition. And all this is because humans are a species that is inherently bad at mathematics.
You may not believe it, all the mathematical talents each of us are born with are just the addition of numbers within 3. Scientists experiment with babies who are a few months old, show him an apple, then cover the apple with a screen, and put another apple behind the screen, the baby will realize that there should be two apples behind the screen. An apple; if the screen is removed and there is still an apple, the baby will be very confused. Another experiment is to test the counting ability of human beings: scientists tested people from different countries and races all over the world, and asked people to count how many black dots there are on a piece of white paper at the fastest speed. The experiment found that if the number of black dots is less than 3, the counting time is almost the same, and the brain can report the answer without thinking, and the correct rate is close to full marks; once the number is more than 5 or 6, the counting time is 1 As the number of black dots increases, counting is often wrong, because the brain cannot directly perceive the number of more than 4, and can only add one by one to find the sum.
These experiments show that only the addition of integers less than 3 such as 1+1=2 is our innate intuition and can be self-evident. There are many interesting circumstantial evidences in linguistics: numbers in Oracle and Roman alphabet, the first 3 numbers are almost the same, 1 is one bar, 2 is two bars, 3 is three bars, 4 and above are different up. The reason can be imagined: if 7 is 7 bars or 7 dots, no one seems to be able to recognize it at a glance! For another example, "three times" in ancient Chinese actually means N times. "Three visits to thatched cottage" does not mean that Liu Bei really convinced Zhuge Liang after only three appointments. In the eyes of the ancients who haven't turned on the math skills, more than 3 are innumerable, and it can only be called "several". If there was a mathematician in ancient times 10,000 years ago, his daily mathematical research work would probably be to count how many lamb legs are left in the hole, and then count how many mouths there are to eat tonight. If there are more numbers, you have to break your fingers or even your toes, so the decimal and binary systems were born. The numbers in French are now a mixture of decimal and twenty. For example, 70 is not called seven 10s, but three 20s plus 10.
Since human beings are born to only do addition within 3, where do the rest of the mathematical world, such as square root numbers and calculus, come from? That's right, these are all deduction based on 1+1=2! If you don't believe me, follow me:
Multiplication is shorthand for "continuous addition". For example, 2x3 is equal to the addition of three 2s; subtraction is adding a negative number; division is "continuous subtraction". The four arithmetic operations of addition, subtraction, multiplication, and division are actually just four different forms of addition. Since multiplication is addition, "serial multiplication" such as squares and cubes, and "serial division" such as square roots and cube roots are also additions. Even the so-called "advanced mathematics" such as calculus, in the final analysis, is addition, because integration is to add infinite numbers together to sum, and differentiation is the inverse operation of integration. From addition, subtraction, multiplication and division to square root and even calculus, everything comes from the most basic addition. Conversely, one who knows how to add can do all calculations. A computer is a classmate who can only do addition. There is only an adder in the CPU, and there are no subtractors and dividers. A simple circuit built with 88 transistors can achieve four-digit addition, and there is no mystery in the CPU's completion of complex operations such as calculus, which relies on calculating billions of additions per second!
When we pulled back, there was a topic that said: If a sprinter races against a tortoise, it can be strictly proved mathematically that the athlete will never catch up with the tortoise, and the fast one cannot run slower than the slow one.
How to prove it? You see, we let the tortoise run 10 meters first, and then the athlete chases forward; when the athlete chases 10 meters away, the tortoise crawls forward a few more steps during this period, for example, 1 meter; the athlete runs another 1 meter, Catch up with the back of the tortoise, and then the tortoise will move forward again... and so on, endlessly. Although the athlete is much faster than the tortoise, he has to run infinite distances and spend infinite amounts of time. time, so he can never catch up with the tortoise.
This question baffled me at the time. According to my rich experience of being at the bottom of sports since I was a child, I believe from the bottom of my heart that athletes can catch up with turtles in a few seconds; but on the other hand, I feel that what is said in the book makes sense, and I am speechless . The sophistry of "athletes can't catch up with turtles" is definitely wrong, but where is it wrong? I took this question to the teacher, but the teacher didn't say why.
I didn't know until later that this weird question turned out to be "Zeno's Paradox", which was the famous work of the ancient Greek philosopher Zeno 2,500 years ago. Moreover, I am not the only one who fell into Uncle Zeno's pit, but also a group of big bulls in ancient Greece, including Aristotle and Archimedes. But when I learned limit summation and calculus, this problem was easily solved.
What is wrong with "Zeno's Paradox"? In a word: the sum of an infinite number of numbers, the result can be a finite value. According to Zeno's logic, in order to catch up with the tortoise, the athlete must run infinitely many distances, but each distance is getting shorter and shorter, and the time it takes to run each distance is getting smaller and smaller. If these infinite distances and infinite time periods are added together, a convergent series is formed, such as 1 + 1/2 + 1/4 + 1/8 or even an infinite number of additions. The result is not Infinity, but exactly equal to 2. Therefore, there is no contradiction between Zeno’s reasoning and our common sense. Athletes will jump over the tortoise within a limited time and a limited distance.
The ancient Greek masters who were the first to study mathematics stopped in the face of limit problems. It never occurred to them that the sum of infinitely many numbers is not necessarily infinite, it may be a finite value. Even though Archimedes, who was Zeno 200 years later, had written the mathematical formula of infinite series, he failed to pierce this layer of window paper and develop the finite sum into an infinite sum. It was not until two thousand years later that Newton, who was born out of nowhere, took this as a breakthrough and opened up an infinite world.
In Newton's calculus, infinities and infinitesimals were commonplace concepts. In the era before Newton, people only calculated the area of squares and circles, and the area algorithms of polygons, trapezoids, and triangles were all derived from squares. Well now, the integral can calculate the area of any irregular shape. Its principle is to cut this geometric shape into infinitely many filaments. The area of this filament is calculated by multiplying the length by the width, and then the infinitely many The total area can be calculated by adding together the filaments with infinitely small areas. With the concept of infinity and limit, mathematics took a big step forward.
Now some people suspect that Newton may have discovered calculus long ago, but he deliberately did not tell others and kept it as his own secret weapon. It wasn't until his good friend Halley came to him and asked him to prove that the orbits of planets are elliptical, and exhausted all kinds of tricks, he persuaded Newton to publish his proof. Three years later, Newton published a book, using his calculus to clearly prove that under the action of universal gravitation, the trajectory of a planet must be an ellipse, and its speed and period can be accurately calculated. So Halley realized that the comet that had been observed many times in ancient times was actually the same one, and it took 75 years to revolve around the sun. In 1758, Halley passed away long ago, and the comet came to the earth as scheduled, so people called it "Halley's Comet". The orbit of Halley's Comet was perhaps the first application of calculus mathematics.
Now when we mention Halley, we often mistakenly think that he is the discoverer of comets. In fact, his greatest contribution to the world is that he sponsored Newton to publish this book: "Mathematical Principles of Natural Philosophy". This book made Newton regarded as the patriarch of mathematics and physics at the same time. It is no exaggeration to say that if Newton hadn't made his three laws and calculus public, our college majors would only be liberal arts, and the modern technology we depend on would disappear.
Mathematics was originally invented to be used as a tool for calculation. As a result, mathematics broke free from human control, and more and more strange numbers came uninvited. These numbers seemed useless and irrational to the ancients, but mathematicians eventually had to invent them.
What are these weird numbers? For example, 0. Anyone is born with the common sense that 1+1=2, but it was not until 2000 years ago that Indian mathematicians invented the concept of "0". Before this, everyone felt that 0 was unnecessary. If I don't have an apple in my hand, I can just say "I don't have an apple". Who would say "I have 0 apples"? Even the 0 used to represent the carry in numbers is unwilling to use, so 2019 is written as 219. As for what it means, it is up to the reader to understand the context. Negative numbers are even more unreasonable to them. Who can say what "-3 sheep" means?
After thousands of years, people finally accepted the concepts of 0 and negative numbers. They knew that "0 sheep" means that the sheep were all taken away by Big Big Wolf, "-3 sheep" owed 3 sheep to Pharaoh next door, and "1 /4 sheep” is a leg of lamb. However, even weirder numbers appeared again: there is a number that is neither an integer, nor can it be written as a fraction that divides integers.
2500 years ago, the Italian Pythagoras discovered that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse. This is called "Pythagorean theorem" in China, and "Hook three strands, four strings and five" refers to a special case of the Pythagorean theorem, that is, the lengths of the right-angled sides are 3 and 4 respectively, and the length of the hypotenuse is 5. However, if the lengths of the sides are 5 and 6, the length of the hypotenuse is no longer an integer, but the square root of 61. Pythagoras' student Hippasus proved that this kind of number with a root sign is neither an integer nor a fraction, and we now call it an "irrational number". However, the discovery of this classmate Xi made Teacher Bi feel like an enemy, because he believed that the world is made up of integers, and if the integers are not enough, there are fractions formed by divisibility. What the hell is a bastard that's not an integer or a fraction? Come on, throw this rebel into the sea and feed the fish for me! Hippasus, who was the first to discover irrational numbers, was brutally killed by his teacher and classmates.
History is such an irony: Pythagoras did not believe in irrational numbers, and as a result, the Pythagorean theorem he discovered by himself put irrational numbers in front of him alive. Just draw a right triangle with side lengths 5 and 6, and you have a hypotenuse of irrational length. Likewise, imaginary numbers were once the outcasts of mathematicians. An imaginary number is a number whose square is negative, and the imaginary number i is equal to the square root of -1. Mathematicians in the 16th century discovered when solving the cubic equation that the general solution of the cubic equation is meaningful only if the root sign -1 is admitted, otherwise it becomes unsolvable. In this way, people have to compromise again. For the convenience of understanding the equation, let’s admit the existence of this product for the time being, but we have to call it an imaginary number—a fictitious number.
No one expected that 400 years later, this "fictitious number" would actually come in handy. Every equation of quantum mechanics contains an imaginary number, because only it can represent the superposition state of the probability wave of particles in different phases, and these states interfere with each other in an incredible way. With imaginary numbers as the cornerstone of mathematics, modern black technologies such as "Schrödinger's cat", "quantum encrypted communication" and "quantum computer" can emerge as the times require.
Here’s to the crazy numbers:
The misfits, the rebels, the troublemakers.
They are not fond of rules, and have no respect of status quo.
They were opposed, ignored, vilified.
Despite obstacles, they push human race forward.
They make us see things differently.
Some may see them as the crazy numbers.
We see genius.
Because the numbers who are crazy enough to be in the imaginary world
have changed the real one.
The emergence of imaginary numbers suddenly broadened the boundaries of the mathematical world. If the world of real numbers composed of integers, fractions, and irrational numbers is a one-dimensional straight line, then the world of imaginary numbers is a two-dimensional plane. Naturally, someone would think: Since numbers can be two-dimensional, can they also be three-dimensional?
This is the question that the 19th-century mathematician Hamilton spent his entire life trying to answer. Hamilton was a genuine genius. He learned all European languages at the age of ten. He was appointed as the Royal Irish Astronomer before he graduated from undergraduate. He also likes to write poetry in his spare time. However, he spent more than ten years, but failed to find this three-dimensional number: "ternary number". All his efforts are always shattered in front of the same problem: no matter how he defines the operation rules of ternary numbers, he cannot define its division. Until one fateful day, Hamilton finally discovered that ternions cannot exist at all, and the only way to solve the problem is to add another dimension to become quaternions.
Quaternions have three dimensions besides the real dimension, called i, j, and k. For example, a typical quaternion looks like this: 1 + 2i + 3j + 4k. Regardless of its strange appearance, the four arithmetic operations of addition, subtraction, multiplication, and division can be performed between any two quaternions. In a mathematical sense, the logic of quaternions is fully established, the question is what is it for? The purpose of Hamilton's pursuit of ternions is to use them to represent three-dimensional space, but he found that ternions do not exist. So what does quaternion represent? Four-dimensional space? What is this extra fourth dimension?
It was not until Einstein created the theory of relativity 50 years later that people finally understood why space must be four-dimensional, because time is also an integral part of space! The high-speed movement of objects in space will cause the effect of slowing down time, and time and space are in a trade-off relationship. Mathematics predicts the real universe in amazing ways.
When we calculate more and more accurate numbers and rejoice in the progress of science and technology, have you ever thought that mathematics is just a fictional concept in the human mind, but it is surprisingly consistent with the laws of the universe? ! In modern physics, the closeness of the two worlds of mathematics and physics can no longer be described as "coincidence". People can even predict undiscovered laws of nature based on mathematical formulas. The positron predicted by Dirac with his electron equation, the gravitational wave predicted by Einstein with the theory of relativity, and the "God particle" predicted by the standard model of quantum mechanics have all been experimentally confirmed. On the eve of the experiment to verify the theory of relativity "gravity distorts space-time", a reporter asked Einstein: What would you think if the experimental results did not match your theory? The old love replied with a cool gesture: "If so, I will feel sorry for God, he doesn't know how to use such beautiful mathematics!".
Mathematics, the extreme sport of the human brain, is climbing a more dangerous peak, trying to get a glimpse of the whole universe. After the quaternion, it was discovered that all numbers have only one-dimensional real numbers, two-dimensional imaginary numbers, and four forms of quaternions and octonions, and operations such as addition, subtraction, multiplication, and division cannot be performed on other dimensions. Imaginary numbers are used in quantum mechanics, and quaternions are used in relativity, so will octonions correspond to deeper and more essential laws of the universe? Although still in the process of exploration, more and more people believe that all the secrets of the universe can be deduced from this new theory based on octonions, including the existing theory of relativity and the standard model of particles. Just because, this is really a kind of "beautiful" mathematics.
As Einstein said: The most incomprehensible thing about the universe is that it is comprehensible. And the way it is understood by humans is mathematics. Why are reality and mathematics so inseparable? Perhaps it is because, like mathematics, this seemingly complex universe evolved from the most basic rules, just like billions of ordinary water molecules arranged together to form a magnificent wave. The big sound has no sound, but the elephant has no form. In front of the wordless universe, what mathematicians see is the power of nature from the avenue to simplicity. They put aside their worldly fame and wealth to devote themselves to mathematics research, just like extreme sportsmen put down life and death, and walk towards the towering peaks and huge waves with a smile, just to stand at a place that no one can reach and have a glimpse of this beautiful scenery, just to face Say a heartfelt admiration to the universe:
「Isn’t that beautiful?」
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