◇ Mathematical knowledge is the purest logical thinking activity and the most advanced aesthetic embodiment of intelligent vitality. Plinson
History makes people wise, poetry makes people witty, and mathematics makes people careful. bacon
Mathematics is one of the most precious research spirits. -Hua
No subject can clarify the harmony of nature more clearly than mathematics. Carlos
Mathematics is the judge and master of law and theory. Benjamin
Music can inspire or soothe feelings, painting makes people pleasing to the eye, poetry can touch people's hearts, and philosophy makes people.
With wisdom, science can improve material life, but mathematics can give all of the above. . -Klein.
Music can inspire or soothe feelings, painting makes people pleasing to the eye, poetry can touch people's hearts, and philosophy makes people.
With wisdom, science can improve material life, but mathematics can give all of the above. . -Klein.
The essence of mathematics lies in freedom. -Cantor
In the field of mathematics, the art of asking questions is more important than the art of answering questions. -Cantor
No problem can touch people's emotions as deeply as infinity, and few other concepts can stimulate reason to produce rich thoughts as infinity, but no other concepts need clarification as infinity. -Hilbert
Mathematics is an infinite science. -Weil
The problem is the core of mathematics. -halmos
As long as a branch of science can raise a large number of questions, it is full of vitality, and no questions indicate the termination or decline of independent development. -Hilbert
Some beautiful theorems in mathematics have such characteristics: they are easy to be summarized from facts, but their proofs are extremely hidden. -Gauss
Mathematics is the queen of science and number theory is the queen of mathematics-Gauss.
Nature's masterpieces are written in mathematical symbols)-Galileo
Mathematics is a tool, especially suitable for dealing with any abstract concept, and its role in this respect is infinite. So a new physics book, if not a simple description of experimental work, must be a math book in essence. -Dirac
Another reason why mathematics is highly respected is that it is mathematics that provides an unquestionable and reliable guarantee for accurate natural science. Without mathematics, they can't achieve such a degree of reliability. -Einstein
Pure mathematics is essentially a poem of logical thinking. -Einstein
Mathematical science presents one of the most brilliant examples, which shows that pure reasoning can successfully expand people's cognitive fields without the help of experiments. -Kant
A person is like a score, his practical ability is like a numerator, and his evaluation of himself is like a denominator. The larger the denominator, the smaller the value of the fraction. -Tolstoy
Time is a constant, but for diligent people, it is a variable. People who calculate time by "minutes" spend 59 times more time than those who calculate time by "hours".
-Rybakov
◇ Dare to do subtraction in learning, that is, to do subtraction on what our predecessors have solved, and see what problems have not been solved, which need us to explore and solve-Hua.
Some beautiful theorems in mathematics have the following characteristics: they are easy to be summarized from facts, but the proof is extremely hidden. Mathematics is the king of science. Gauss
Mathematics is an infinite science. Herman Weil
In the world of mathematics, what matters is not what we know, but how we know it.
Pythagoras
Only by successfully applying mathematics can a science be truly perfect.
Marx
The scientific level of a country can be measured by the mathematics it consumes.
-Rao
◇A=x+y+z. A stands for success, X stands for hard work, Y stands for correct method, and Z stands for less empty talk.
-Einstein
Genius = 1% inspiration +99% blood and sweat. Edison
Take time to think about what you have done in a day, whether it is "plus sign" or "minus sign". If it is "plus sign", you will make progress. If it is "-",we must learn a lesson and take measures. -dimitrov.
Life should be like a line segment with a beginning and an end; It should not be like a ray, with a beginning and no end.
The trajectory of life is a circle, but you can extend the radius of the circle.
A person should seek the greatest value in a limited living area.
People in their twenties are sharp, people in their thirties are obtuse, people in their forties are straight, and people in their fifties are rounded.
Being friends should be like a vertical line and communicate with each other; Being an opponent should be like parallel lines. Although you come and go, you chase after me and surpass each other.
Mathematical story:
Yes161811. Descartes served in the army and was stationed in a small city in the Netherlands to fill Boleda. One day, when he was walking in the street, he saw a group of people gathered near a sign posting a notice. They were talking excitedly. He approached curiously. But because he couldn't understand Dutch and the Dutch characters on the notice, he asked the people next to him in French. A passer-by who can understand French looked disapprovingly at the young soldier and told him that there was a prize contest to solve mathematical problems. If you want him to translate all the contents of the notice, you need one condition, that is, the soldiers should send him the answers to all the questions in the notice. The Dutch claimed that he was a teacher of physics, medicine and mathematics. Unexpectedly, the next day, Descartes really came to him with the answers to all the questions; What surprised Beckman in particular was that all the answers of the young French soldier were not wrong at all. As a result, the two became good friends, and Descartes became a frequent visitor to Beckman's house.
Descartes began to study mathematics seriously under the guidance of Beckman, who also taught Descartes to learn Dutch. This situation lasted for more than two years, which laid a good foundation for Descartes to create analytic geometry later. Moreover, it is said that the Dutch words that Buick taught Descartes also saved Descartes' life:
Descartes once sailed to France with his servant on a small merchant ship, and the fare was not very expensive. I didn't realize this was a pirate ship. The captain and his deputy thought that Descartes' master and servant were French and didn't understand Dutch, so they negotiated to kill them in Dutch and robbed them of their money. Descartes understood the words of the captain and his deputy, made preparations quietly, finally subdued the captain and returned to France safely.
Eight-year-old Gauss discovered mathematical theorems.
He entered a rural primary school at the age of eight. The teacher who teaches mathematics is from the city. He feels that teaching a few little lynx in remote places is really overqualified. Moreover, he has some prejudices: children from poor families are born fools, and there is no need to teach these stupid children to study hard. If there is an opportunity, they should be punished to add some fun to this boring life.
This day is a depressing day for the math teacher. The students cringed when they saw the teacher's depressed face, knowing that the teacher was going to arrest these students again today and punish them.
"You calculate for me today, from 1 plus 2 plus 3 to 100. Whoever can't figure it out will be punished for not going home for lunch. " The teacher said this, picked up a novel, sat in a chair and read it without saying a word.
The children in the classroom picked up the slate and began to calculate: "1 plus 2 equals 3, 3 plus 3 equals 6, 6 plus 4 equals10 …" Some children added a number to the slate and then erased the result. After adding it, the number is getting bigger and bigger, which is difficult to calculate. Some children's little faces turned red, and some children's palms and foreheads oozed sweat.
Less than half an hour later, little Gauss picked up the slate and stepped forward. "Teacher, is this the answer?"
Without looking up, the teacher waved his thick hand and said, "Go, go back!"! Wrong. " He thought it impossible to have an answer so soon.
But Gauss stood still and put the slate in front of the teacher: "Teacher! I think this answer is correct. "
The math teacher wanted to shout, but when he saw the number written on the slate: 5050, he was surprised because he calculated it himself and got the number of 5050. How did this 8-year-old child get this value so quickly?
Gauss explained a method he discovered, which was used by the ancient Greeks and China people to calculate the sequence1+2+3+…+n. Gauss's discovery made the teacher feel ashamed, and felt that his previous view of being arrogant and belittling poor children was wrong. He also taught seriously in the future, and often bought some math books from the city for his own study and lent them to Gauss. With his encouragement, Gauss later did some important research in mathematics.
Xiaooula zhigai sheepfold
Euler is a famous mathematician in the history of mathematics. He has made outstanding achievements in several branches of mathematics, such as number theory, geometry, astronomical mathematics and calculus. However, this great mathematician was not liked by teachers at all when he was a child. He is a student expelled from school.
Things are caused by stars. At that time, little Euler was studying in a missionary school. Once, he asked the teacher how many stars there were in the sky. The teacher is a believer in theology. He doesn't know how many stars there are in the sky, and the Bible doesn't answer. In fact, there are countless stars in the sky, which are infinite. There are thousands of stars visible to the naked eye. Without pretending to understand, the teacher replied to Euler, "It doesn't matter how many stars there are in the sky, as long as you know that the stars in the sky are inlaid by God."
Euler felt very strange: "The sky is so big and so high, and there is no escalator on the ground. How did God embed the stars on the screen one by one? " God himself put them in the sky one by one. Why did he forget the number of stars? Could God be too careless?
He asked the teacher a question in his heart, and the teacher was confused again, blushed and didn't know how to answer. A sudden anger rose in the teacher's heart, not only because a child who just went to school asked the teacher such a question, so that the teacher could not step down, but more importantly, the teacher regarded God above everything else. Little Euler blames God for not remembering the number of stars. The implication is that he doubts almighty god. In the teacher's view, this is a serious problem.
In Euler's time, there was absolutely no doubt about God. People can only be slaves of ideas, and they are absolutely not allowed to think freely. Little Euler didn't "keep in line" with the church and God, so the teacher told him to leave school and go home. However, in little Euler's mind, the sacred aura of God disappeared. God is a loser, he thought. Why can't he even remember the stars in the sky? He thought, God is a dictator, and even asking questions has become a crime. He thinks that God may be a guy made up by others and does not exist at all.
After returning home, he helped his father herd sheep and became a shepherd boy. He read a book while herding sheep. Among the books he read, there are many math books.
Dad's flock increased gradually, reaching 100. The original sheepfold was a little small, so my father decided to build a new sheepfold. He measured a rectangular piece of land with a ruler, 40 meters long and 15 meters wide. He calculated that the area is exactly 600 square meters, with an average of 6 square meters per sheep. When he was ready to start construction, he found that his materials were only enough for the fence of 100 meters, which was not enough at all. If you want to enclose a sheepfold with a length of 40m and a width of 15m, its circumference is110m (15+15+40+40 =1/kloc-0). If we want to build it according to the original plan, my father feels very embarrassed. If the area is reduced, the area of each sheep is less than 6 square meters.
Little Euler told his father that there was no need to shrink the sheepfold and not to worry that the territory of each sheep would be smaller than originally planned. He has an idea. Father didn't believe that little Euler would have a way, so he ignored him. Little Euler was in a hurry and said loudly, only moving a little stake in the sheepfold.
Father shook his head and thought, "How can there be such a cheap thing in the world?" However, little Euler insisted that he would be able to kill two birds with one stone. The father finally agreed to let his son try.
Little Euler saw that his father agreed, stood up and ran to the sheepfold to start work. He shortened the original side length of 40 meters to 25 meters centered on the stake. The father was anxious and said, "How can that be done? Then how to do it? This sheepfold is too small, too small. " Little Euler didn't answer, so he ran to the other side and extended the original side length of15m, and increased it by10m to 25m. In this way, the original planned sheepfold has become a square with a side length of 25 meters. Then, little Euler confidently said to his father, "Now, the fence is enough and the area is enough."
Father built a fence according to the sheepfold designed by little Euler. 100 meter fence is really enough, no more, no less, all used up. The area is enough, a little bigger. Father felt very happy. Children are smarter than themselves, they really use their brains, and they will definitely have a bright future.
Father thinks it's a pity to let such a clever child herd sheep. Later, he tried to let little Euler know the great mathematician Bernoulli. On the recommendation of mathematicians, little Euler became a college student in university of basel on 1720. This year, little Euler 13 was the youngest student in this university.
Interesting math problem:
1. Two boys each ride a bicycle, starting from two places 20 miles apart (1 mile +0.6093 km) and riding in a straight line. At the moment they set off, a fly on the handlebar of one bicycle began to fly straight to another bicycle. As soon as it touched the handlebar of another bicycle, it immediately turned around and flew back. The fly flew back and forth, between the handlebars of two bicycles, until the two bicycles met. If every bicycle runs at a constant speed of 10 miles per hour and flies fly at a constant speed of 15 miles per hour, how many miles will flies fly?
answer
The speed of each bicycle is 10 miles per hour, and the two will meet at the midpoint of the distance of 2O miles after 1 hour. The speed of a fly is 15 miles per hour, so in 1 hour, it always flies 15 miles.
Many people try to solve this problem in a complicated way. They calculate the first distance between the handlebars of two bicycles, then return the distance, and so on, and calculate those shorter and shorter distances. But this will involve the so-called infinite series summation, which is very complicated advanced mathematics. It is said that at a cocktail party, someone asked John? John von neumann (1903 ~ 1957) is one of the greatest mathematicians in the 20th century. ) Put forward this question, he thought for a moment, and then gave the correct answer. The questioner seems a little depressed. He explained that most mathematicians always ignore the simple method to solve this problem and adopt the complex method of summation of infinite series.
Von Neumann had a surprised look on his face. "However, I use the method of summation of infinite series," he explained.
2. A fisherman, wearing a big straw hat, sat in a rowboat and fished in the river. The speed of the river is 3 miles per hour, and so is his rowing boat. "I must row a few miles upstream," he said to himself. "The fish here don't want to take the bait!"
Just as he started rowing upstream, a gust of wind blew his straw hat into the water beside the boat. However, our fisherman didn't notice that his straw hat was lost and rowed upstream. He didn't realize this until he rowed the boat five miles away from the straw hat. So he immediately turned around and rowed downstream, and finally caught up with his straw hat drifting in the water.
In calm water, fishermen always row at a speed of 5 miles per hour. When he rowed upstream or downstream, he kept the speed constant. Of course, this is not his speed relative to the river bank. For example, when he paddles upstream at a speed of 5 miles per hour, the river will drag him downstream at a speed of 3 miles per hour, so his speed relative to the river bank is only 2 miles per hour; When he paddles downstream, his paddle speed will interact with the flow rate of the river, making his speed relative to the river bank 8 miles per hour.
If the fisherman lost his straw hat at 2 pm, when did he get it back?
answer
Because the velocity of the river has the same influence on rowing boats and straw hats, we can completely ignore the velocity of the river when solving this interesting problem. Although the river is flowing and the bank remains motionless, we can imagine that the river is completely static and the bank is moving. As far as rowing boats and straw hats are concerned, this assumption is no different from the above situation.
Since the fisherman rowed five miles after leaving the straw hat, he certainly rowed five miles back to the straw hat. Therefore, compared with rivers, he always paddles 10 miles. The fisherman rowed at a speed of 5 miles per hour relative to the river, so he must have rowed 65,438+00 miles in 2 hours. So he found the straw hat that fell into the water at 4 pm.
This situation is similar to the calculation of the speed and distance of objects on the earth's surface. Although the earth rotates in space, this motion has the same effect on all objects on its surface, so most problems about speed and distance can be completely ignored.
3. An airplane flies from city A to city B, and then returns to city A. In the absence of wind, the average ground speed (relative ground speed) of the whole round-trip flight is 100 mph. Suppose there is a persistent strong wind blowing from city A to city B. If the engine speed is exactly the same as usual during the whole round-trip flight, what effect will this wind have on the average ground speed of the round-trip flight?
Mr. White argued, "This wind will not affect the average ground speed at all. In the process of flying from City A to City B, strong winds will accelerate the plane, but in the process of returning, strong winds will slow down the speed of the plane by the same amount. " "That seems reasonable," Mr. Brown agreed, "but if the wind speed is 100 miles per hour. The plane will fly from city A to city B at a speed of 200 miles per hour, but the speed will be zero when it returns! The plane can't fly back at all! " Can you explain this seemingly contradictory phenomenon?
answer
Mr. White said that the wind increases the speed of the plane in one direction by the same amount as it decreases the speed of the plane in the other direction. That's right. But he said that the wind had no effect on the average ground speed of the whole round-trip flight, which was wrong.
Mr. White's mistake is that he didn't consider the time taken by the plane at these two speeds.
It takes much longer to return against the wind than with the wind. In this way, it takes more time to fly when the ground speed is slow, so the average ground speed of round-trip flight is lower than when there is no wind.
The stronger the wind, the more the average ground speed drops. When the wind speed is equal to or exceeds the speed of the plane, the average ground speed of the round-trip flight becomes zero, because the plane cannot fly back.
4. Sunzi Suanjing is one of the top ten famous arithmetical classics in the early Tang Dynasty, and it is an arithmetic textbook. It has three volumes. The first volume describes the system of counting, the rules of multiplication and division, and the middle volume illustrates the method of calculating scores and Kaiping with examples, which are all important materials for understanding the ancient calculation in China. The second book collects some arithmetic problems, and the problem of "chickens and rabbits in the same cage" is one of them. The original question is as follows: let pheasant (chicken) rabbits be locked together, with 35 heads above and 94 feet below.
Male rabbit geometry?
The solution of the original book is; Let the number of heads be a and the number of feet be b, then b/2-a is the number of rabbits and a-(b/2-a) is the number of pheasants. This solution is really great. When solving this problem, the original book probably adopted the method of equation.
Let x be the pheasant number and y the rabbit number, then there is
x+y=b，2x+4y=a
Get a solution
y=b/2-a，
x=a-(b/2-a)
According to this set of formulas, it is easy to get the answer to the original question: 12 rabbits, 22 pheasants.
Let's try to run a hotel with 80 suites and see how knowledge becomes wealth.
According to the survey, if we set the daily rent as 160 yuan, we can be full; And every time the rent goes up in 20 yuan, three guests will be lost. Daily expenses for services, maintenance, etc. Each occupied room is calculated in 40 yuan.
Question: How can we set the price to be the most profitable?
A: The daily rent is 360 yuan.
Although 200 yuan was higher than the full price, we lost 30 guests, but the remaining 50 guests still brought us 360*50= 18000 yuan. After deducting 40*50=2000 yuan for 50 rooms, the daily net profit is 16000 yuan. When the customer is full, the net profit is only 160*80-40*80=9600 yuan.
Of course, the so-called "learned through investigation" market was actually invented by myself, so I entered the market at my own risk.