18th century, a great German scientist Gauss (karl friedrich, 1777- 1855) was born. He is the most outstanding astronomer and mathematician of our time. Gauss, known as the "prince of mathematics", is one of the founders of modern mathematics, and can be juxtaposed with Archimedes, Newton and Euler.
Gauss showed outstanding mathematical talent when he was young. When he was still in primary school, one day the teacher asked the students to calculate the following formula: 1+2+3+…+ 100.
For primary school students, this is not a simple addition operation. However, Gauss can easily write the correct answer 5050.
What method does Gauss use to calculate the result so quickly? It turned out that he found that 1 and 100 were added first to get101; 99 plus 2 is also101; If you keep adding it, * * * has 50 10 1, then the result of this formula is 10 1 50 = 5050.
In this way, Gauss skillfully solved the problem by using the algorithm. Do you understand the mystery?
In fact, we can use the formula to calculate the sum of the first n positive integers, that is, 1+2+3+…+n, and this formula is also a general term of triangular numbers.
Intelligent Pyramid ── Thales
Thales (about 625-574 BC), born in Miletus on the southwest coast of Asia Minor, was an ancient Greek mathematician, astronomer and philosopher. Thales is a shrewd businessman. Because he foresaw a bumper harvest of olive oil, he earned a lot of money by renting and selling equipment for making olive oil, which enabled him to have enough money for scientific research and travel.
Thales likes to travel around. According to legend, when Pharaoh traveled in Egypt, he ordered the priests to measure the height of the pyramid (Pharaoh's tomb), which worried the priests. In order to help the priests solve their difficulties, Thales used an ingenious method to measure the height of the pyramid.
Thales erected a wooden pillar beside the pyramid. When the length of the wooden column shadow is equal to the length of the wooden column, the height of the pyramid can be obtained by measuring the length of the pyramid shadow. This shows Thales' talent in mathematics and science.
Pythagoras and trigonometric numbers
Speaking of Pythagoras (about 5565438 BC+0-479 BC), we are most familiar with the Pythagorean Theorem. However, Pythagoras' passion is not geometry.
Pythagoras was an ancient Greek mathematician. He believes that every number has a unique personality, good and evil. He even thinks that 10 is a perfect number. This is because 10 is the sum of the first four positive integers 1, 2, 3 and 4, and it is a triangular number. Musically, if a string with a length of 1 unit is tightened, a tone do can be generated. If the length of the string is changed to the ratio of these four positive integers: sum, the tonic produced is fa, so and do have a high average.
Pythagoras founded a school called Pythagoras School. This school is very well organized and has a strong religious color. They think numbers are the root of everything. They study numbers not for practical application, but to reveal the eternal truth of the universe through their understanding of numbers. Unfortunately, many research results have been lost because the school strictly abides by the principle of confidentiality.
Archimedes, mathematician of Syracuse
Archimedes (about 287 BC-2 BC12 BC) was born in Syracuse, Greece. His father was an astronomer. Archimedes received a good education from an early age. When he was young, he went to Alexandria to study mathematics.
The volume of the crown
Once, a goldsmith in Black Luo Rang, King of Syracuse, made a crown out of pure gold, but he suspected that the goldsmith had hidden some gold. The goldsmith denied it and confirmed that the weight of the crown was equal to that given by the king. The king was at a loss, so he asked Archimedes for help.
Archimedes pondered the solution day and night. He knows that even if different materials have the same weight, their volumes are different, so we can judge whether the crown is made of pure gold from its volume, but there is no way to get its volume.
Once, when Archimedes was taking a bath in the bathtub, he saw the water slowly flowing out of the plate, so he realized that the volume of the crown could be calculated by drainage method. If you put the crown in a plate filled with water, the volume of water discharged is the volume of the crown. In this way, Archimedes solved the problem for the king and proved that the goldsmith did mix silver into the crown.
Don't spoil my photo.
"Don't spoil my painting"-this is Archimedes' last sentence.
In 2 12 AD, the Romans invaded Syracuse. According to legend, Archimedes was studying mathematics at that time, and a Roman soldier broke into Archimedes' home and stepped on geometric figures. Archimedes didn't pay attention to who the other person was. He shouted "Don't spoil my picture" and was killed by the soldiers.
Master of Surveying ── Helen
Heron of Alexandria (about 1 century) was born in Egypt and was an ancient Greek mathematician, mechanic, mechanic and surveyor. She once taught mathematics and physics in Alexandria, a famous academic research city in the Roman Empire. Helen attaches great importance to the practical application of mathematics, which can be learned from his works Geodesy, Geometry and Volume Solution. For hundreds of years, ancient people have been using geodesy. In addition, he also used to annotate and supplement Euclid's Elements of Geometry (about 330-275 BC).
Helen is famous for solving geometric measurement problems. He gave many formulas for calculating the area and solid volume of plane graphics, such as the area calculation method from regular triangle to regular dodecagon. In geodesy, he even gave the famous triangle area formula-Helen formula.
In addition, Helen also applied his theory to mechanical design, and wrote books such as Mechanics, Grabbed Gun and Gun Design. At the same time, he is also the designer of water clocks, measuring instruments and cranes. It can be seen that he is a genius who applies mathematics to life.
Rooster 5 yuan 3 hens 5 yuan 2, sold together 10 yuan 5, lost?
A few days ago, Bayi didn't succeed in being clever, and he lost seven silver rings for nothing. It was so painful. He has been hurting people. How can he swallow this tone? It's not like he swaggered around the market. I wonder who will have bad luck again today?
"Sell chickens, 3 cocks 5 yuan, 2 hens 5 yuan, come and buy!" Following the hawking, Master Bayi came to the chicken beach, only to see his thief-like eyes turn, and his plan came to mind. "Hey, old man, how many cocks do you have here? How many hens? " "There are 30 each." The old man selling chicken answered with a trembling voice.
How old are they?
"What are you up to, Bill?" The professor said attentively. At this time, his friend was drinking the rest of the coffee in one breath and stood up to leave. "I want to take three girls to travel by bus!" Bill replied.
The professor smiled: "So that's it!" How old are the three beautiful women? Bill thought for a moment and said, "Multiply their ages to get 2450, but they are exactly twice as old as you." ".
The professor shook his head and said, "Very clever, but their age is still a problem." Bill is still here. He added, "Yes, I forgot to say that I am at least one year younger than the oldest." This makes everything clear!
Goldbach's Conjecture
Goldbach is a German mathematician.
From 1729 to 1764, Goldbach kept correspondence with Euler for 35 years.
In the letter 1742 to Euler on June 7th, Goldbach put forward a proposition. He wrote:
"My question is this:
Take any odd number, such as 77, which can be written as the sum of three prime numbers:
77=53+ 17+7;
Take an odd number, such as 46 1,
46 1=449+7+5,
It is also the sum of three prime numbers, and 46 1 can also be written as 257+ 199+5, or the sum of three prime numbers. In this way, I found that any odd number greater than 7 is the sum of three prime numbers.
The oldest and most interesting problem in mathematics
In seven houses, each house has seven cats; No matter which of the seven cats can catch seven mice; Each of these seven mice eats Three wheat; If each ear of wheat can peel off seven grains of wheat, how many houses, cats, mice, ears of wheat and grains of wheat add up?
Answer:
The total is 19607.
There are 7 rooms in the house, with 7x7 = 49 cats, 7X7X7=343 mice, 7x7x7 = 240 1 ear wheat and 7x7 = 16807 grains of wheat. All this means
Archimedes' tombstone
Compared with heroes' monuments or tombstones, only mathematicians' epitaphs are the simplest. Their tombstones are often just engraved with a number or a number. These shapes and figures show their persistent pursuit and brilliant achievements all their lives.
The ancient Greek mathematician Archimedes (287 BC-2 BC12 BC) had such a tombstone. On his tombstone, there is a cylinder engraved with a ball. The diameter of the ball is exactly equal to the height of the cylinder.
This figure, called "equilateral cylinder", expresses Archimedes' discovery that "the volume and surface area of a sphere are equal to two-thirds of the volume and surface area of its circumscribed cylinder". It is not difficult to prove, so students might as well give it a try.
Honeycomb conjecture
Devlin, a Canadian science journalist, wrote in the Globe and Mail that after 1600 years of hard work, mathematicians finally proved that bees are the most efficient builders in the world.
Pepos, an ancient Greek mathematician in the 4th century, pointed out that the beautiful shape of beehives is the representative of the most efficient labor in nature. He guessed that the honeycomb with hexagonal cross section was built by bees with minimal beeswax. His conjecture is called the "hive conjecture", but no one can prove it.
Hale, a mathematician at the University of Michigan, announced that he had cracked the conjecture. Honeycomb is a very sophisticated construction project. When bees build nests, young and middle-aged worker bees are responsible for secreting flaky fresh beeswax, each with only one needle, and other worker bees are responsible for carefully placing these beeswax in a certain position to form a vertical hexagonal column. The thickness and error of each beeswax partition wall are very small. The width of the six partition walls is exactly the same, and the included angle between walls is exactly 120 degrees, forming a perfect geometric figure. People have always wondered why bees don't make their nests into triangles, squares or other shapes. Why is the partition flat, not curved? Although the beehive is a three-dimensional building, each beehive is a hexagonal cylinder, and the total area of beeswax wall is only related to the cross section of the beehive. This leads to a mathematical problem, that is, to find the plane figure with the largest area and the smallest perimeter.
Mathematical metaphor
Many celebrities like to use mathematics to compare things, often out of humor and wit, which gives people a very deep impression.
Parallel lines
Time will pierce the gorgeous feelings of youth, carve parallel lines on a beauty's forehead and eat rare treasures. Beauty is born, and nothing can escape its sickle-Shakespeare.
Interesting math story
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.
Bears buy fish.
Bell's mother is ill. In order to earn money to treat her, Bear gets up before dawn every day to go fishing in the river and sell fish in the market as soon as possible.
One day, as soon as Little Bear set up the fish stall, the fox, the black dog and the old wolf came. When the bear saw a customer coming, he hurriedly called, "Buy fish? I just caught this fish. How fresh! " The fox turned the fish over and asked, "How much is a kilogram of such fresh fish?" "The bear grinned." Cheap, four yuan a kilo. The old wolf shook his head: "I am old and my teeth are falling out." I just want to buy some fish. "Bear's face is reluctant:" I'll sell you the fish. Who will I sell the fish head and tail to? The fox wagged his tail and said, "Yes, no one wants to buy the leftovers, but Uncle Wolf has bad teeth and can only eat some fish. To tell you the truth, Black Dog and I are very good to each other. If you buy fish heads and tails, won't you help Uncle Wolf and Brother Xiong? "The bear clapped his hands, but he hesitated." Good, but at what cost? "The fox rolled his eyes and replied," 2 yuan is 1 kg, fish head and fish tail are 1 yuan is 1 kg. Isn't it 4 yuan 1kg? "The bear drew a picture on the ground with a stick, and then patted his thigh:" OK, let's start! " "The four of them started work together, and in a short time, the head, tail and body of the fish were separated. Once the bear weighed, the body of the fish was 35 kilograms of 70 yuan; Fish head 15 Jin 15 yuan, fish tail 10 Jin 10 yuan. The wolf, the fox and the black dog quickly ran to the Woods with the fish, matched the head, the body and the tail, and then divided them equally.
On the way home, Bear thought: According to 4 yuan 1 kg, I should sell 60 kg of fish to 240 yuan, but now I only sell 95 yuan ... Bear can't figure it out. Do you know what this is about?
Interesting math story
The lame fox sells watermelons at a loss and has no money to buy food. He was so hungry that he kept walking.
The old cow came over and asked, "Fox, what's wrong with you?"
The fox looked at the old cow and said, "Hungry, I haven't eaten anything seriously for two or three days!" " "
The old cow said solemnly, "If you want to have food, you have to take part in labor!" " "Say that finish, the old cow went to work.
"Hum, labor? How tired this labor is! " The fox rolled his eyes and said, "Well, I have a good idea."
The fox limped to the wild boar's house. The wild boar family has a big basket with a lot of corn in it. The basket is covered with thick cloth. The fox said, "Brother Wild Boar, I heard that there is a lot of corn in this basket. Can you tell me how many? "
"Secret!" The wild boar replied grumpily.
"Ha ha, there can't be any secrets in front of my clever fox!" The fox said confidently, "When I debut, you can count. I can not only know how many corn cobs are in your basket, but also know how old you are. "
"Really?" The wild boar felt incredible.
The fox coughed twice and said, "multiply the number of corn cobs in your basket by 2, add 5, multiply it by 50, add your age and subtract 250. Tell me the number."
The wild boar squatted on the ground for a long time, and finally said, "take 1506."
The fox immediately said, "You have 15 corn cobs in your basket. You are 6 years old. "
The wild boar touched his forehead and thought, yes, there are 15 corn cobs in the basket. The wild boar touched the back of his head and thought he was 6 years old this year.
"Oh, my God!" The wild boar admires the fox from the heart. He asked the fox, "How do you know?"
"Come on! Your calculation result is 1506. The leftmost two digits 15 are the number of the corn cob; The rightmost number 6 is your age. "
"You are great!" The wild boar hugged the fox and kissed it.
"It doesn't matter whether it's great or not. The important thing is to get me a meal, with wine and meat! " The fox looks proud.
After a while, the wild boar brought the fox stewed rabbit meat, steamed chicken, boiled old corn and two bottles of good wine. The fox ate and drank, and took four corn cobs before he left.
The wild boar advertised everywhere that the lame fox had a clever plan. Lingling, the little monkey, told the wild boar, "You have been taken in by the fox!" " "Wild boar don't believe it.
The monkey said, "Look at the formula (2×15+5 )× 50+6-250 =15×100+250+6-250 =1500+6 =/kloc. The number on the corn cob (15) was written by myself. Multiplied by 100, it became a number of thousands and hundreds. At the age of 6, it was also written by myself and became a single digit. In doing so, the two figures are separated and clear at a glance. "
"What a lame fox!" The wild boar rushed out quickly, caught up with the lame fox, grabbed the corn cob and gave the fox a hard knock on the head with each corn cob. Great, the lame fox has added four big bags to his head!
Mathematical short stories
Recommended by Chen Junjiao
Today, the integer kingdom is very lively, because Zero King's birthday is today, and today is the New Year's Day. Double happiness came to the door, and all the civil and military officials came to celebrate.
I saw Zero King perched on the throne, and the following two rows of teams lined up. A line starts with-1, followed by -2, -3, -4 ... They are shorter than each other. The other line starts with Commander 1, followed by 2, 3, 4, 5 ..... one is higher than the other, and there is no end in sight.
Three shots were fired and the celebration began. Suddenly, a round little thing emerged from under the king's throne. 1 The commander drew his sword, took a few steps and shouted, "Who's here?" The little thing said slowly, "Why, I don't even know. I tell you, I am the famous decimal point. " What can I do for you? 1 The commander always speaks so forcefully. "I'm here to attend the zero king celebration. Please help me arrange to join the team! " Without waiting for 1 commander to respond, Zero King said, "No, look at the long queue outside the palace. Civil servants begin with prime minister-1, and military officers begin with commander 1. There is no place for you. " The decimal point pleaded, "You see I'm so small, just give me your seat!" " ""no, you'd better leave quickly. " "Hum! Toast does not eat or drink. " The decimal point changed his face and snapped, "I want you to make a big change in order!" " "
King Zero was furious and shouted, "Grab this little thing and make a big number." Listen, a tall man comes from outside the palace. It's 9700000097 million. He shouted, "Decimal point, where can you escape 1?" Decimals are not afraid. It jumped on the throne, picked up King Zero, pushed it in front of 97 million people and stood in front of King Zero. Bang, 97 million yuan, which is higher than the mountain, has become 0.097 yuan, which is shorter than the chair.
King Zero was frightened and shouted, "Whoever can master the decimal point will be named monarch." I saw a tumbler figure coming from outside the palace, and 8 said, "You can't catch the decimal point, you can only outwit it it." "Well," the decimal point said happily, "I'd like to see how you can be smarter than me. 8 said: "decimal point, I just witnessed your skill, which is really extraordinary." But you can only make one number smaller. I wonder if you have any other skills? "
The decimal point smiled: "There is a negative number, only -47, and the decimal point will soon be in the middle of 4 and 7. -47 immediately grew a lot and became -4.7. " According to negative numbers, the smaller the absolute value, the larger the value. I won't let negative numbers get bigger. ""well "
Then, 8 said, "In my opinion, there is only one person who is not afraid of decimal points." King Zero leaned down. "Who is this man?" It is you. Me? Why am I not afraid? ""Because you are neither positive nor negative, 0.0 is still 0! The decimal spell doesn't work on you. "As soon as the decimal point hears zero, the king can turn himself in. He ran away with a squeak before he finished eight sentences.
No matter how small the mistake is, it is also wrong.
1 On August 23, 967, the Soviet Union1spacecraft suddenly had a vicious accident when it returned to the atmosphere-the parachute could not be opened. After studying it, the central leadership of the Soviet Union decided to broadcast the accident live to the whole country. When the TV announcer announced in a heavy tone that the spaceship would crash in two hours and the audience would witness 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, everything is fine, don't worry! " 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 today on Alliance 1 is because a decimal point was ignored during ground inspection ... "
Time is running out. There are only seven minutes before the spaceship crashes. 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".
Price of decimal point Author: Anonymous article Source: China Foundation Network
1 On August 23, 967, the former Soviet Union1spacecraft suddenly had a vicious accident when it returned to the atmosphere-the deceleration parachute could not be opened. After studying it, the central leadership of the former Soviet Union decided to broadcast the accident live to the whole country. When the TV announcer announced in a heavy tone that the spaceship would crash in two hours and the audience would witness 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, everything is fine, don't worry! " At this time, komarov's daughter also appeared on the TV screen. She is only 12 years old. Komashov 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 today on Alliance 1 is because a decimal point was ignored during ground inspection ... "
Time goes by, only seven minutes before the spaceship crashes. Komalov waved to the national television audience and said, "Fellow citizens, please allow me to say goodbye to you in this vast space."
This is a soul-stirring farewell ceremony. Komalov is gone forever, leaving behind his eternal love for his relatives and the motherland. But what is even more shocking is what he said to his daughter. It warns people not to be careless about life, otherwise, even a trivial matter will make you pay a heavy or even irreparable price.
Enlightenment from Zu Chongzhi.
Author: Capital Normal University ... Article Source: Mathematics Network Arrangement ~c, CngeL0
There is an asteroid in the vast night sky and a crater on the back of the distant moon. They are all named after an ancient scientist in China. He is Zu Chongzhi (429-500), an outstanding mathematician, astronomer and mechanical manufacturing expert in the Southern and Northern Dynasties.
Zu Chongzhi was born in a family that studied astronomical calendars for generations. Influenced by the environment, he was very interested in mathematics and astronomy since he was a child. In the Book of Song Dynasty, Zu Chongzhi said: "Although I am not so sharp and stupid, I am still good at mathematics and have a thorough knowledge of ancient and modern times. Later, I studied Xia Dian, felt the amount, Zhou Zhengshuo, and added salt to test ... "When Zu Chongzhi was a child, he collected and read a large number of mathematical documents of his predecessors, and made a thorough and systematic study of these materials. He insisted on personally checking and verifying every step of calculation, not being bound by the achievements of predecessors, correcting his own mistakes and combining his own understanding and creation, which greatly promoted the development of ancient mathematics in China in the following three aspects;
One is the calculation of pi. He calculated 3.1415926 < < 3.1415927 as the density. The range of values and the calculation of encryption rate are more than 1000 years ahead of foreign countries.
The second is the calculation of ball volume. Zu Chongzhi and his son Zuheng found a formula to calculate the volume of a ball. The "Zuheng principle" used in this formula and "If it is evenly matched, the product cannot be different", that is, the contour is 9? X8H 1
The volumes of two geometric bodies with equal cross-sectional areas must be equal. Until 1 100 years later, the Italian mathematician cavalieri put forward an axiom with similar significance.
The third is to annotate "Nine Chapters of Arithmetic" and write "Grade Book". As a teaching material of mathematics education in Tang Dynasty, Seal Script is known as "a scholar can't study its depth", but this precious ancient book has long been lost.
Zu Chongzhi's achievements in mathematics made China not only catch up with the Greeks in some aspects of mathematics, but even lead them by a thousand years. It has been 1,500 years since Zu Chongzhi died. What does Zu Chongzhi's scientific achievements inspire us middle school students?
Numbers confuse our lives.
Author: anonymous article Source: World Science and Technology Report
When the price of computers dropped by 2000 yuan compared with the previous year and the price of meat rose by 3 yuan, did your family's living expenses decrease or increase? If you deposit your money in the bank, can the bank interest be offset by rising prices? These questions can be answered by statistics.
At the end of the Qin Dynasty, Chen Sheng and Guangwu shouted the slogan "It is better to have seeds than princes", and there is a famous couplet "Heroes have suffered a lot since ancient times, and dude is not as good as a great man", but the statistics give us different answers. Thousands of years of imperial examinations show that only 50% of the Jinshi come from rural areas, and the rest are from officials and nobles. At that time, more than 90% of the population in China lived in rural areas, including small and medium-sized landlord families. In contrast, the proportion of scholars who really come from peasant families is even less. Even in today's so-called democratic America, you can see this phenomenon. George W. Bush, the son of President Bush, is also the president, and the Kennedy family is actually a political aristocratic family. Although democracy can give everyone an equal opportunity to compete on the surface, statistics tell us that people born in bureaucratic families actually have more opportunities to enter the upper class.
This shows that statistics can often correct our inherent intuition about social phenomena.
Intensive store knowledge
Author: anonymous article Source: Reprinted
The shape of floor tiles is often square and rectangular, and we have also seen regular hexagonal floor tiles. Whether it is a square, rectangular or hexagonal floor tile, the middle of a piece of ground can be paved without gaps or overlaps, that is, densely paved. What other shapes of graphics can be densely laid on the ground? When students think about this problem, they always experiment with the help of the pictures they draw and draw conclusions through actual observation.