Interesting knowledge of mathematics

Number theory part:

1, there is no maximum prime number. Euclid gave a beautiful and simple proof.

2. Goldbach conjecture: Any even number can be expressed as the sum of two prime numbers. Chen Jingrun's achievement is that any even number can be expressed as the sum of the products of one prime number and no more than two prime numbers.

3. Fermat's last theorem: n power of x+n power of y = n power of z, and n> has no integer solution at 2 places. Euler proofs 3 and 4, 1995 were proved by British mathematician andrew wiles.

Topology part:

1. The relationship among points, faces and edges of a polyhedron: fixed point+number of faces = number of edges +2, which was proposed by Descartes and proved by Euler, also known as euler theorem.

2. euler theorem's inference: There may be only five regular polyhedrons, namely regular tetrahedron, regular octahedron, regular hexahedron, regular icosahedron and regular dodecahedron.

3. Turn the space upside down, the left-handed object can be changed into the right-handed, and through Klein bottle simulation, a good mental gymnastics,

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2. Little knowledge of mathematics

This is an interesting common sense of mathematics, and it is also good to use it in mathematics newspapers.

People call 12345679 "Leak 8". This "number without 8" has many surprising characteristics, such as multiplying by multiples of 9, and the product is actually composed of the same number. People call this "uniform". For example:12345679 * 9 =1111/kloc-0. 27 = 333333333 ... 1 2345679 * 81= 999999 These are all 9 times of1multiplied by 9.

And 99, 108, 1 17 to 17 1. The final answer is:12345679 * 99 =1222212345679 *108 =13333333212345677. 444444443 ..... Paradox: (1) Russell Paradox One day, the barber in Saville Village put up a sign: All men in the village who don't cut their hair themselves will be cut by me.

So someone asked him, "Who will cut your hair?" The barber was speechless at once. 1874, the German mathematician Cantor founded the theory of * * * *, which quickly penetrated into most branches and became their foundation.

By the end of19th century, almost all mathematics was based on * * * * theory. At this time, a series of contradictory results appeared in the theory of * * *.

Especially in 1902, Russell put forward the paradox reflected in The Barber's Story, which is extremely simple and easy to understand. In this way, the foundation of mathematics has been shaken passively, which is the so-called third "mathematical crisis".

Since then, in order to overcome these paradoxes, mathematicians have done a lot of research work, produced a lot of new achievements, and brought about a revolution in mathematical concepts. (2) liar paradox: "What I said is a lie."

This paradox put forward by the Greek mathematician Euclid in the fourth century BC still puzzles mathematicians and logicians. This is the famous liar paradox.

A similar paradox first appeared in the 6th century BC, and Epimini, a Crete philosopher, once said, "All Cretes are lying." There is also a very similar sentence in China's ancient Mo Jing: "Words are contradictory, and their words are also."

It means: it is wrong to think that everything is wrong, because it is a sentence. The liar paradox takes many forms. For example, write the following two sentences on the same piece of paper: The next sentence is a lie.

The last sentence is true. What is more interesting is the following dialogue.

A said to B, "What you want to say next is' no', right? Please answer with' yes' or' no'! " This is another example. There was a devout believer who kept saying in his speech that God was omnipotent and omnipotent.

A passerby asked, "Can God make a stone that he can't lift?" 2.*** Numbers In life, we often use the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Do you know who invented these numbers? These digital symbols were first invented by ancient Indians, and then spread to * * *, and then from * * * to Europe. Europeans mistakenly think that it was invented by * * * people, so it is called "* * * number". Because it has been circulating for many years, people still call them * * *.

Now, the number * * * has become a universal digital symbol all over the world.

3. Interesting math essay

Interesting mathematical story 1, butterfly effect meteorologist Lorenz put forward a paper entitled "Will butterflies flap their wings to cause tornadoes in taxonomic groups?" This paper discusses that if the initial condition of a system is a little worse, its result will be very unstable. He called this phenomenon "the butterfly effect".

Just like we roll the dice twice, no matter how deliberately we roll, the physical phenomena and points thrown twice are not necessarily the same. Why did Lorenz write this paper? This story happened in the winter of 196 1 2008. He operated the meteorological computer in the office as usual.

Usually, he only needs to input meteorological data such as temperature, humidity and air pressure, and the computer will calculate the possible meteorological data at the next moment according to the built-in three differential equations, thus simulating the meteorological change map. On this day, Lorenz wanted to know more about the subsequent changes of a record. He re-entered the meteorological data at a certain moment into the computer, so that the computer could calculate more subsequent results.

At that time, the speed of computer processing data was not fast enough, so he had time to have a cup of coffee and chat with his friends for a while before the results came out. An hour later, the result came out, but he was dumbfounded.

Compared with the original information, the original data is similar, and the later data is more different, just like two different pieces of information. The problem is not the computer, but the data he entered is 0.0005438+027. These subtle differences make a world of difference.

So it is impossible to accurately predict the weather for a long time. References:

Cao Cao's Gourd (Volume II)-Yuan Zhe Science Education Foundation II. The mathematical "genius" hive in animals is a strict hexagonal cylinder with a flat hexagonal opening at one end and a closed hexagonal diamond bottom at the other end, which consists of three identical diamonds.

The rhombic obtuse angle of the chassis is 109 degrees 28 minutes, and all acute angles are 70 degrees 32 minutes, which is both firm and material-saving. The honeycomb wall thickness is 0.073 mm, and the error is very small.

Red-crowned cranes always move in groups, forming a "human" shape. The angle of the herringbone is 1 10 degrees.

More accurate calculation also shows that half the angle of the herringbone-that is, the angle between each side and the direction of the crane group is 54 degrees, 44 minutes and 8 seconds! And the angle of diamond crystal is exactly 54 degrees, 44 minutes and 8 seconds! Is it a coincidence or some "tacit understanding" of nature? The spider's "gossip" net is a complex and beautiful octagonal geometric pattern, and it is difficult for people to draw a symmetrical pattern similar to a spider's net even with the compass of a ruler. In winter, when a cat sleeps, it always hugs its body into a ball. There is also mathematics in it, because the shape of the ball minimizes the surface area of the body, so it emits the least heat.

The real "genius" of mathematics is coral. Coral writes a "calendar" on its body, and "draws" 365 stripes on its wall every year, apparently one a day.

Strangely, paleontologists found that corals 350 million years ago "painted" 400 watercolors every year. Astronomers tell us that at that time, the earth only had 2 1.9 hours a day, not 365 days a year, but 400 days.

(Life Times) 3. Every piece of paper in Mobius tape has two sides and a closed curved edge. If there is a piece of paper with one side and only one side, is it possible for an ant to reach another point from any point on the paper without crossing the edge? In fact, it is possible. Just twist a piece of paper tape in half and stick both ends on it. This is the German mathematician Mobius (M? Beus. A.F 1790- 1868) was found in 1858. Since then, that kind of belt has been named after him, called Mobius belt.

With this toy, a branch of mathematical topology can flourish. 4. Mathematician's Will * * * The will of mathematician Hua Lazimi, when his wife was pregnant with their first child.

"If my dear wife gives birth to a son for me, my son will inherit two thirds of the inheritance and my wife will get one third; If it is a girl, my wife will inherit two-thirds of the inheritance and my daughter will get one-third. " .

Unfortunately, the mathematician died before the child was born. What happened after that made everyone more troubled. His wife gave birth to twins, and the problem happened in his will.

How to follow the mathematician's will and divide the inheritance among wife, son and daughter? 5. Matching Games One of the most common matching games is that two people play together. Put some matches on the table first, and two people take turns to take them. Each time, there can be some restrictions on the number of competitions, stipulating that the person who wins the final competition wins. Rule 1: How can we win if the number of competitions we participate in at one time is limited to at least one and at most three? For example, there are n= 15 matches on the table. Party A and Party B take turns to take it, and Party A takes it first. How should Party A lead them to win? In order to get the last one, A must leave zero matches for B at the end, so A can't leave 1 or 2 or 3 in the round before the last step, otherwise B can win all of them.

If there are four games left, then B can't win them all, so no matter how many games B wins (1 or 2 or 3), A can win all the remaining games. Similarly, if there are 8 matches left on the table for B to take, no matter how B takes them, A can leave 4 matches after this round, and finally A must win.

As can be seen from the above analysis, A only needs to make the matching numbers in Table 4, 8, 12, 16. Let B get it, and A will be a shoo-in.

Therefore, if the original number of matches on the table is 15, A should take three matches. (∫ 15-3 = 12) What if the original matching number on the table is 18? Then A should take 2 pieces first (∵ 18-2= 16).

Rule 2: If the number of matches taken at one time is limited to 1 4, how can we win? Principle: If Party A takes it first, then every time Party A takes it, it must leave a multiple of 5 matches for Party B.. General rule: There are n matches, and you can take 1 to k matches at a time, so the number of matches left after each take of A must be a multiple of k+ 1.

Rule 3: How to limit the number of matches taken at one time to some discontinuous numbers, such as 1, 3, 7? Analysis: 1, 3, 7 are all odd numbers. Since the target is 0, and 0 is an even number, then the first person who takes A must make the matching number on the table even, because B is among the even matching numbers, it is impossible to get 0 after matching with 1, 3, 7, but if.

4. Who has a little knowledge of mathematics?

Yang Hui triangle is a triangular numerical table arranged by numbers. The general form is as follows:11113314641151. 1 561721353521kloc-0/........................................................................... Yanghui triangle's most essential feature is that its two hypotenuses are all composed of the number1,and the rest are equal to the sum of its upper two numbers.

In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one.

Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram.

And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Now we are required to output such a table through programming.

At the same time, this is also the law of the quadratic coefficient of polynomial (a+b) n after bracket opening, that is, 0 (a+b) 0 (0ncr0)1(a+b)1ncr0) (1ncr65438). (2 NCR 1)(2 NCR 2)3(a+b)^3(3 NCR 0)(3 NCR 1)(3 NCR 2)(3 NCR 3)。 . .

. .

. Therefore, the Y term of X layer in Yang Hui Triangle is directly (y nCr x), and it is not difficult for us to get that the sum of all terms in X layer is 2 x (that is, when A and B in (A+B) X are both 1) [Y X refers to the X power of Y; In fact, ancient mathematicians in China were far ahead in many important mathematical fields.

The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one. Yang Hui was born in Hangzhou in the Northern Song Dynasty.

In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram. And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way.

Specific usage will be taught in the teaching content. Abroad, it is also called Pascal Triangle. There is also a short story: (1) is wrong, and it is a thousand miles away. 1 On August 23, 967, the Soviet Union1spacecraft suddenly suffered a serious accident when it returned to the atmosphere-the parachute could not be opened.

After studying, the Soviet Union decided to broadcast the accident live to the whole country. When the announcer of the TV station announced in a heavy tone that the spaceship would crash in two hours and the audience would witness the news of the martyrdom of astronaut Vladimir Komarov, the whole country was immediately shocked and people were immersed in great grief.

On TV, the audience saw the calm image of astronaut komarov. He smiled and said to his mother, "Mom, I can clearly see your image here, including every gray hair on your head. Can you see me clearly? " "Yes, you see very clearly.

Son, mom, everything is fine, you can rest assured! "At this time, komarov's daughter also appeared on the TV screen. She is only 12 years old. Komarov said, "Daughter, don't cry. "

"I don't cry ..." My daughter broke down in tears, but she fought back her grief and said, "Dad, you are a Soviet hero. I want to tell you that the hero's daughter will live like a hero! " Komarov told her daughter, "When you study, you should take every decimal point seriously. What happened to Soyuz-1 today was because a decimal point was ignored in the ground inspection ... "One minute passed, only seven minutes before the spaceship crashed.

Komalov waved to the national television audience and said, "Fellow citizens, please allow me to say goodbye to you in this vast space." Even a decimal point error will lead to a tragic farewell that can never be remedied.

Julius Caesar of ancient Rome famously said, "In war, great things are often the result of small things." In China's epigram, it's probably "a short trip makes a long regret".

(2) Chen Jingrun, a mathematician triggered by a story, is a famous mathematician. He has made great contributions to overcoming Goldbach's conjecture and founded the famous "Chen Theorem", so many people affectionately call him "the prince of mathematics". But who would have thought that his achievement originated from a story?

1937, diligent Chen Jingrun was admitted to Huaying College in Fuzhou. At this time, during the period of War of Resistance against Japanese Aggression, Professor Shen Yuan, director of the Department of Aeronautical Engineering in Tsinghua University, returned to Fujian to attend the funeral, unwilling to stay in his hometown because of the war. Several universities got the news and wanted to invite Professor Shen to give lectures. He declined the invitation.

As he is an alumnus of Huaying, he came to this middle school to teach mathematics to his classmates in order to report to his alma mater. One day, Teacher Shen Yuan told us a story in math class: "A Frenchman discovered an interesting phenomenon 200 years ago: 6 = 3+3, 8 = 5+3, 10 = 5+5, 12 = 5+7, 28 = 5+23.

Every even number greater than 4 can be expressed as the sum of two odd numbers. Because this conclusion has not been proved, it is still a guess.

Euler said: Although I can't prove it, I am sure this conclusion is correct. It is like a beautiful light ring, shining with dazzling brilliance in front of us not far away.

..... "Chen Jingrun stare eyes, absorbed. From then on, Chen Jingrun became interested in this wonderful question.

In his spare time, he likes going to the library. He not only read the counseling books in middle schools, but also eagerly read the textbooks of mathematics and physics courses in these universities. Therefore, he got the nickname "bookworm".

Interest is the first teacher. It is such a mathematical story that aroused Chen Jingrun's interest and his diligence and made him a great mathematician.

(3) People who are crazy about science, because of endless research, often get some logical but absurd results (called "paradox"), and many great mathematicians take an evasive attitude because they are afraid of falling into it. During the period of 1874- 1876, Cantor, a young German mathematician less than 30 years old, declared war on the mysterious infinity.

With hard sweat, he successfully proved that points on a straight line can correspond to points on a plane one by one, and can also correspond to points in space one by one. In this way, the point on the line segment 1 cm long seems to be in the Pacific Ocean.

5. Interesting mathematical knowledge in life

1. A garment worker can produce 4 coats or 7 pairs of trousers per person every day, and one coat and one pair of trousers make a suit.

At present, 66 workers produce, how many sets of clothes can you produce at most every day? Xiao Wang has three stamp albums, one fifth of all stamps are in the first one, n divided by 8(N is a non-zero natural number) is in the second one, and the remaining 39 stamps are in the third one. How many stamps does Xiao Wang have? Xiao Ming looked at his report card and predicted that if the next math test is 100, the total average score will be 9 1. If he gets 80 points in the next exam, the overall average score will be 86 points. How many times did Xiao Ming take the math statistics exam? 1 There are x workers producing tops. If it is 4x=7*(66-x), then x=42, then 4*42= 168 sets of clothes can be produced in one day. 2. It has X stamps, and its x/5+N/8+39=x is simplified to 4x/5-N/8=39. 39 is an odd number, so n is the odd tail of 8. Let N=(2t+ 1)*8 be 4x/5-(2t+1) = 39x = (100+5t)/2 be an even number, and let t=2w be x = (.

At this time, N=32w+83 had x test scores, and now the average score is A, so there is (xa+100)/(x+1) = 91(xa+80)/(x+1) = 80.

6. Collect interesting stories about mathematics.

1. The symbols "+"and "-"were first used by a German 500 years ago.

At that time, they didn't mean "plus" or "minus". It was not until more than 300 years ago that it was officially used to mean "addition" and "subtraction".

2. "Tangram" is a kind of tangram in ancient China, which was made up of seven pieces and made into a large square thin plate, with various patterns. Later, it spread abroad and was called "Tangtu".

"Tangram" has spread to today and become a kind of intellectual toy that people like. It is said that as early as 4,000 to 5,000 years ago, our ancestors used a dripping instrument to keep time. This instrument is called lettering.

More than 300 years ago, an English mathematician first used the multiplication symbol "*". Because multiplication is a special addition, he marked the plus sign obliquely.

In 46 BC, Julius Caesar, commander-in-chief of Rome, designated the calendar. Because he was born in July, in order to show his greatness, he decided to change July to "Julian month", and all single months were stipulated as 3 1 day, and bimonthly was 30 days.

In this way, there is one more day in a year. February was the month when prisoners were executed in ancient Rome. In order to reduce the number of people executed, 1 day was reduced in February to 29 days. 6. Xiao Fang is a carpenter, but he is arrogant. One day, the master asked him, "The table has four corners. I cut off one, how many are left? " Xiao Fang said 4- 1=3, three.

The master told him that there were five sevenths. About 1500 years ago, European mathematicians did not know how to use "0". They use Roman numerals.

Roman numerals are symbols representing numbers, which are combined to represent different numbers according to certain rules. When using this number, the number "0" is not needed.

At that time, a scholar of the Roman Empire discovered the symbol "0" from Indian notation. He found it very convenient to use "0" for mathematical operation, and he was very happy. He also introduced the Indian method of "0" to everyone.

After a while, it was known by the pope at that time. At that time, it was the Middle Ages in Europe. The power of the church was very strong, and the power of the pope far exceeded that of the emperor.

The Pope was very angry. He rebuked that the sacred number was created by God, and there was no such monster as "0" in the number created by God. Anyone who wants to introduce now is blasphemous! So the Pope ordered the scholar to be arrested and tortured, and his ten fingers were tightly clamped with a clamp, so that his hand was disabled and he could no longer write with a pen. In this way, "0" was banned by the ignorant and cruel Pope.

However, although the use of "0" is forbidden, Roman mathematicians still use "0" secretly in mathematical research regardless of the ban, and still make a lot of mathematical contributions with "0". Later, "0" was finally widely used in Europe, but Roman numerals were gradually eliminated.

8. Children, do you know the story of Gauss, a mathematical genius, when he was a child? When Gauss was in elementary school, once after the teacher taught addition, because the teacher wanted to have a rest, he came up with a topic for students to calculate. The title is: 1+2+3+. ..+97+98+99+ 100 = ? The teacher is thinking, now the children must start class! I used this as an excuse to go out, but Gauss stopped me! ! It turns out that Gauss has worked it out. Little friend, do you know how he did it? Gauss told everyone how he worked it out: add 1 to 100, add 100 to 1, and add two lines, that is 1+2+3+4+.

..+96+97+98+99+ 100 100+99+98+97+96+ 。 ..+4+3+2+ 1 = 10 1+ 10 1+ 10 1+ 。

... =' class1' >+10/+1+1+10 1 * * There are one hundred10/kloc-0 ... In daily life, mathematics is everywhere, such as: buying and selling vegetables, calculating how much money ... 9. The following is a short story, which is a story between numbers. One day, when the digital cards were having lunch together, the youngest one spoke.

Brother 0 said, "Let's take some photos together. What do you think? " 0' s brothers and sisters said in unison, "Good." Brother 8 said, "Brother 0' s idea is really good. I will be a good person once. Shall I provide camera and film for Brother 8? " Old four said, "starling, yes, it's just a little troublesome." Better use my digital camera. That's settled. "

So, they got busy. Finally, they took pictures for them and immediately sent the digital camera to the print shop. The computer sister tried to ask them for money, but who paid for it? They stared at each other one after another. This is what the computer sister said, "A ***5 yuan, a * * * eleven brothers and sisters, how much does one person pay on average?" Among the eleven of them, Lao Liu is the cleverest, and this time he is the first to work out the result. Do you know how it is worked out? 10. One day, the Tang Priest told his disciples Wukong, Bajie and Friar Sand to go to Huaguoshan to pick peaches. Before long, the three disciples returned happily after picking peaches.

Tang Priest and his disciples asked: How many peaches did each of you pick? Bajie said with a silly smile, Master, let me test you. Each of us took the same amount of money. There are less than 65,438+000 peaches in my basket. If we count three peaches, there are 1 peach left in the end.

Do the math. How many did each of us choose? Friar Sand said mysteriously, Master, I will test you, too. If there are four peaches in my basket, there is 1 left in the end.

Do the math. How many did each of us choose? Wukong smiled and said, Master, I'll test you, too. If there are five peaches in my basket, there will be 1 in the end.

Do the math. How many shall we each choose? Tang Priest quickly said the number of peaches he had picked. Do you know how many peaches they picked? .

7. Collect 20 math skills

1。

The vertex angles are equal. 2. Pi is an irrational number.

3。 The sum of the internal angles of a triangle is 180 degrees 4.

The sum of the inner angles of a polygon is (number of sides -2)* 180 degrees 5. The sum of the outer angles of a polygon is equal to 360 degrees 6.

The image of a linear function is a straight line. 7。

The image of the proportional function is a straight line passing through the origin. 8。

The image of inverse proportional function is a hyperbola. 9。

The image of a quadratic function is a parabola. 10。

Same radix power multiplication, constant radix, exponential addition. 1 1。

Two parallel lines are cut by a third straight line and have the same angle. 12。

Two parallel lines are cut by a third line, and their inner angles are equal. 13。

Two parallel lines intersect with the third straight line, which complement each other. 14。

The three median lines of a triangle intersect at a point, which is called the center of gravity. 15。

The bisectors of the angles of a triangle intersect at a point, which is called the heart. 16。

The three heights of three sides of a triangle intersect at a point, which is called the center of gravity. 17。

The perpendicular lines of the three sides of a triangle intersect at a point, which is called the epicenter. 18。

Two triangles with the same base and height have the same area. 19。

1+2+3+……+n =( 1+n)* n/2 20 .Sin90= 1，Cos90=0，Sin0=0，Cos0= 1 .