Mathematical reasoning story

Gauss's grandfather was a farmer, and his father was engaged in various fields besides gardening.

All kinds of handyman, such as berm, builder and so on. Because of poverty, his father didn't suffer.

What kind of education have you received?

My mother got married at the age of 34 and gave birth to Gauss at the age of 35. She is a stone.

The craftsman's daughter has a very clever younger brother, whose dexterity and brains are famous for weaving silk locally.

Gauss's uncle, Hand, takes care of little Gauss and educates him every chance he gets.

Teach him something he knows. And the father can be said to be a "lout", think

Only strength can make money, and learning is useless to the poor.

In his later years, Gauss likes to tell his little grandson stories about his childhood. he said

He had learned to calculate before he could speak.

When he was less than three years old, one day he watched his father calculating the jobs under his jurisdiction.

People's weekly salary. Father was mumbling to count, finally sighed and finally counted the money.

Come out.

When my father finished reading the money and was about to write it down, a small voice came from around him: "Dad!

The calculation is wrong. The money should be like this. 」

Father was surprised to calculate it again, and sure enough, the number that Little Goss said was right, which is the strange place.

No one has taught Gauss how to calculate, but little Gauss usually relies on observation, not on adults.

Unconsciously, he learned to calculate by himself.

Another famous story can also show that Gauss had fast computing ability at an early age.

Force. When he was still in primary school, one day, the arithmetic teacher asked the whole class to work out the following formula.

The following equation:

1 + 2 + 3 + 4 + ....+ 98 + 99 + 100 = ?

Soon after the teacher asked the question, Gauss wrote the answer on his small stone board.

In case 5050, there are other children dizzy, but they still can't figure it out. Finally, only gauss.

The answer is correct.

Originally it was1+100 =101.

2 + 99 = 10 1

3 + 98 = 10 1

.

.

.

50 + 5 1 = 10 1

The first item and the second item are added together to get 50 pairs, all of which are 10 1.

That is, 10 1 × 50 = 5050.

News: the current formula

It means 1+2+...+n

Goss's family is very poor. After supper in winter evening, his father will go to Gauss.

Sleeping in bed can save fuel and lamp oil. Gauss likes reading very much. He often

He took a bundle of turnips to his attic. He hollowed them out and stuffed them with coarse cotton.

The coiled wick uses some grease as candle oil, so it gives off a faint light here.

Under the lamp, concentrate on reading. When he felt tired and cold, he went to bed.

Go to sleep.

Gauss's arithmetic teacher has a bad attitude towards his students. He often thinks he is.

Teaching in the backcountry is a rare talent, and now he is happy to find a "child prodigy"

. But soon he felt ashamed, because he didn't know much about mathematics, and it was impossible to judge the height correctly.

What can Alice do to help?

He went to town and bought a math book for Gauss, who was very happy.

Learn this book with the help of a teacher who is almost ten years older than him. This child

Establish a deep affection with that teenager, and they spend a lot of time discussing.

Things.

Gauss discovered the generality of binomial theorem (x+y )n at the age of eleven.

Case, where n can be a positive integer and a negative integer, or a positive and negative fraction. When he was a pupil.

When I was young, I paid attention to infinite problems.

One day, Gauss was walking home, absorbed in reading while walking. number

Unconsciously walked into the garden of Braunschweig Palace, when Braunschweig

The Duchess of Swick saw the child like reading so much that she talked to him.

She found that he fully understood the profound content of the book he read.

The duchess went back to report to the duke, who also heard that she was under his jurisdiction.

There is a story about a clever boy in our territory, so we sent someone to call Gauss to the palace.

Duke Ferdinand also likes this shy child very much.

Appreciating his talent, he decided to give him financial assistance and give him a chance to receive higher education.

Education, Ferdinand's care for Gauss is beneficial, otherwise Gauss's father is opposed.

For children who study too much, he always thinks that it is more important to work to earn money than to do some math research.

Useful, so how can Gauss become a useful person?

Gauss's school career

With the help of Duke Ferdinand, 15-year-old Gauss entered a famous

Junior college (equivalent to high school and university). He studied ancient times there.

And began to study advanced mathematics.

He devoted himself to reading the works of famous European mathematicians Newton, Euler and Lagrange.

Works. He especially praised Newton's work and soon mastered Newton's differential product.

Subtheory

1795 10 In June, he left his hometown college to study in G? ttingen.

study The University of G? ttingen is very famous in Germany, and its rich mathematical collection attracts Gauss.

. Many foreign students also go there to study language, theology, law or medicine. this is

A city with a strong academic atmosphere.

Gauss doesn't know which department to study at this time, the language department or the mathematics department. such as

From a practical point of view, it is not easy to find a life after learning mathematics.

But on the eve of his eighteenth birthday, a new discovery in mathematics made him make up his mind.

Decided to study mathematics for life. This discovery is very important in the history of mathematics.

We know that when n ≥ 3, regular N polygons refer to those polygons with equal sides.

N-sided polygons with the same internal angles.

Greek mathematicians have long known to draw plus 3 with compasses and scale-free rulers.

Four, five and fifteen pentagons. But for more than two thousand years, no one knew.

How to construct regular eleven sides, thirteen sides, fourteen sides and seventeen sides with rulers and compasses

Edge shape.

Gauss, who is not yet eighteen, found that a regular N-polygon can be composed of a ruler and a square.

Compasses are drawn if and only if n is one of the following two forms:

k = 0. 1.2， ...

17th century, the French mathematician Fermat thought this formula.

Prime numbers are given at k = 0, 1, 2, 3, ... (In fact, only F0, F 1, F2 and F4 have been determined at present.

Is a prime number, F5 is not).

Gauss solved geometric problems for more than two thousand years by algebraic method, and found that.

Practice of regular heptagon ruler and compass. He was so excited that he decided.

I have been studying mathematics all my life. It is said that he also expressed the hope that he could engrave it on his tombstone after his death.

A regular heptagon to commemorate his most important mathematical discovery as a teenager.

1799, Gauss submitted his doctoral thesis, which proved that algebra is a heavy one.

Important Theorem: Any unary algebraic equation has roots. This result is mathematically called "generation"

The basic theorem of numbers ".

In fact, many mathematicians in Gauss think that this result has been given.

Prove, but not strictly. Gauss was the first to give strict.

Accurate proof, Gauss thinks this theorem is very important in his life.

One * * * four different proofs. Gauss has no money to publish his paper. It doesn't matter.

Duke Ferdinand gave him money for printing.

At the age of twenty, Gauss wrote in his diary that he had many mathematical ideas.

In my mind, because the time is uncertain, I can only record a small part. Fortunately, he put the research

The results of the research were written into a book called Arithmetic Research, which was published at the age of 24.

This book is written in Latin. It originally had eight chapters, but due to lack of money, it had to be printed in seven chapters.

This book can be said to be the first systematic book on number theory, which Gauss introduced for the first time.

I "this concept.

Zhi Dou Zhu Bajie

It is said that Tang Priest and his disciples came to guo jia cun after learning Buddhist scriptures in the West, and were warmly welcomed by the villagers. Everyone regards them as heroes of exorcism, not only taking photos with them as a souvenir, but also inviting them to be guests at home.

Facing the hospitality of the villagers, the teachers and apprentices felt very sorry, and helped them harvest crops, plow fields and rake fields whenever they had the chance. In the first few days, the pig worked hard, but after a few days, he became lazy again. He thinks it's too hard to work like this, and the master is more comfortable. As long as you sit and preach and recite Buddha, you will have everything. In fact, the master is nothing. If it weren't for the Monkey King's critical eye and skill, I'm afraid I couldn't even go to the Western Heaven, let alone learn from the Buddhist scriptures. If I have such an apprentice, I can do something, and then, haha, I can enjoy my retirement.

So Bajie began to talk. Within a few days, he recruited nine apprentices, and he named them: Xiao Yi Huan, Xiao Er Huan ... Xiao Jiu Huan. It stands to reason that Bajie should concentrate on practicing and teaching his disciples now. However, he still did not change his bad habits, and often took his disciples out to eat and drink, which made the people complain. People are thinking about the good things they have done for everyone, and no one is embarrassed to complain to Wukong. In this way, Bajie people are even more fearless, eating five or six hundred steamed buns at a time, and the people are almost boiling.

There is a girl named Ganoderma lucidum in the neighboring village. She is clever and kind, and often uses her wisdom to fight the wicked skillfully. Hearing this, she decided to punish Bajie. When she came to guo jia cun to open a restaurant, Bajie heard the news. Lingzhi girl pretended to be surprised and said, "Master Wuneng, it's a great honor for you to come to my restaurant. Come to my place for dinner in the future and don't go anywhere else. " She paused and said, "Here is a round table specially prepared for you. You ten people sit in a different order every time. When you complete all the orders, I will provide you with meals for free. But before that, every time you eat a meal, you should do a good deed for a villager in the village. What do you think? " Hearing this tempting suggestion, Bajie was so excited that they repeatedly said yes. So they always eat according to the agreed conditions and write down the seats. A few years later, new orders are still emerging, and Bajie is puzzled and has to ask Wukong for advice. Wukong couldn't help laughing when he heard this. "You idiot, you can't count such a simple bill, and you still want to pick it up cheaply. You'll never get this free meal. " "Can't we get it after twenty or thirty years?" Wukong said, "Then I'll work out this account for you. Let's start with a simple number. Suppose there are three people eating. We gave them the serial numbers of 1, 2, 3, and there were six sequences, namely 123, 132, 2 13, 23 1 3 12, 3265433. If four people eat a piece of metal, the first person will sit still, and the seats of the other three people will change six times. When four people sit still as the first person in turn, the overall arrangement order is 6× 4 = 24. According to the same method, it can be inferred that if five people go to eat, there will be 24× 5 = 120 arrangements ... 10 people will have 3628800 different arrangements. Because you have to eat three meals a day, you can calculate the number of days to eat with 3628800÷3: 1209600 days, which is nearly 3320 years. Think about it, can this free leek be eaten? "

After Wukong's calculation, Bajie suddenly understood the intention of Ganoderma lucidum girl and couldn't help feeling ashamed. From then on, Bajie often took his migrant brother to help the villagers. They regained people's love.

Countermeasures to win

During the Warring States Period, Qi Weiwang and Tian Ji raced, and Qi Weiwang and Tian Ji each had three good horses: getting on, winning and dismounting. The race is divided into three times, and thousands of dollars are bet on each horse race. Because the horsepower of the two horses is almost the same, and Qi Weiwang's horse is better than Tian Ji's, most people think that Tian Ji will lose. However, Tian Ji took the advice of his disciple Sun Bin (a famous strategist) and dismounted Qi Weiwang, Ma Zhong of Qi Weiwang and Qi Weiwang. As a result, Tian Ji beat Qi Weiwang 2-/kloc-0-and got a daughter. This is an example of China's ancient substitution game theory to solve problems.

Here is a game played by two people: take turns to report numbers, and the number reported cannot exceed 8 (nor can it be 0). Add up the figures reported by two people, and whoever reports more figures will win if the total is 88. If you were allowed to count first, how many times should you count first to win?

Analysis: Because everyone reports at least 1 and at most 8 at a time, someone reports and another person will find a number, so the sum of this number and a reported number is 9. According to the rules, whoever counts and makes the sum 88 wins, so it can be inferred that whoever counts and makes the sum 79 (= 88-9) wins. 88 = 9× 9+7, and so on. Whoever counts 16 wins. Furthermore, whoever reports 7 first will win. Therefore, the winning strategy of the first whistleblower is: report 7 first, and then if the other party reports K( 1≤K≤8), you report (9-K). In this way, you will win if you quote 10.

When does the snail climb up the well?

A snail accidentally fell into a dry well. It lay at the bottom of the well and began to cry. Mass (

Lai) Toad crawled over and whispered to the snail, "Don't cry, little brother! It's no use crying. The shaft wall is too high. If you fall here, you can only live here. I have been here for many years, and I haven't seen the sun for a long time, let alone want to eat swan meat! " The snail looked at the old and ugly toad and thought, "What a beautiful world outside the well! I can never live in a dark and cold well like it! " The snail said to the toad, "Uncle toad, I can't live here. I have to climb up! "! How deep is this well? ""Ha ha ha ..., are you kidding! The depth of this well is10m. How can you climb up with such a heavy shell at a young age? " "I am not afraid of suffering, not afraid of being tired. I can always climb out after a while every day! " The next day, the snail ate and drank enough and began to climb the wall. It kept climbing and finally climbed 5 meters at night. The snail was very happy and thought, "At this rate, I can climb up tomorrow night." Thinking about it, it fell asleep unconsciously. In the morning, the snail was awakened by a purr. At first glance, it turned out that Uncle Tu was still sleeping. It was surprised: "How come I am so close to the bottom of the well?" It turned out that this snail slipped 4 meters from the borehole wall after falling asleep. The snail sighed, gritted his teeth and began to climb again. At night, it climbed another 5 meters, but at night the snail slipped another 4 meters. Climb and climb, and finally the strong snail climbed up the well platform. Children, can you guess how many days it will take a snail to climb the drilling platform?