1962 65438+India issues commemorative stamps on February 22nd. This stamp is in memory of India.

"National Treasure" Sirinivasa? 75th anniversary of Srinivasa Ramanujan's birth.

Has been released.

Ramanujiang was born in a poor Brahmin family during the decline of South India and had no university education.

Through self-study and hard study of mathematics, he later became an internationally famous mathematician.

Mathematicians, born in poverty, can be lonely without learning mathematics.

Not many people have found some in-depth results in their work. He didn't know the truth until he was twenty-seven years old.

Under the guidance of a mathematician, his talent suddenly appeared in the sky like a comet, dazzling. unfortunately

Lung disease took his life, and he died quietly at the age of 33.

He was born in Tamir,1887 65438+February 22nd, and his father was a small shop assistant in a cloth shop. small

Sometimes he spends most of his time at his grandmother's house. He likes thinking since he was a child. He once asked his teacher.

The distance between the shining constellations in the sky and the length of the earth's equator. I began to learn mathematics at the age of twelve.

I once asked my senior classmates, "What is the highest truth of mathematics?" At that time, his classmates told him, "Bida.

Gauss theorem (called quotient theorem in China) can be regarded as a representative, which has attracted his attention to several issues.

What interest?

One day, a teacher said, "Thirty fruits are divided equally among thirty people, and everyone gets one."

. The same fourteen fruits are distributed to fourteen people on average, and each person gets a fruit. "From here, teacher.

The conclusion is that any number divided by itself is one. Rama Jiang felt wrong and immediately stood up and asked:

"Does everyone have it?" At this time, the wonderful nature of numbers caught his attention, which was also pitiful.

Not much. At this time, he made his own research on the properties of arithmetic and proportional series.

At the age of thirteen, his senior classmates lent him a copy of Lionel's Trigonometry.

Some schools adopt this book as a high school curriculum, and its Chinese translation is called Trigonometry of Dragons. He learned it quickly.

The exercises in the whole book have been solved. The next year, he got the infinite series expansion of sine and cosine functions.

Later, I learned that this was the famous Euler formula. I was a little disappointed, so I put my own draft results,

Secretly put it on the roof of the room.

When he 15 years old, his friends lent him two thick books entitled The Application of Pure Numbers, written by an Englishman Carl.

Summary of the basic results of mathematics. This book is very boring, listing algebra, micro

Six thousand theorems and formulas of integral, trigonometry and analytic geometry. This book is a good book for him,

He proved some of these theorems himself, and the foundation of his research in the future was given by this book.

1930 entered the government college in my hometown. Because of poverty and excellent entrance examination results, he was awarded.

He won a scholarship, but in college, he was too absorbed in his math to pay attention to other subjects.

As a result, I failed the annual exam and lost my scholarship. 1906 in the second year, he transferred to another college to study and

Take 1907' s "The First Examination of Liberal Arts". Yes, it failed again.

From 1907 to 19 10, he lived outside and couldn't find any job, sometimes filling in for his friends.

Study is in exchange for food. During this period, he studied the Rubik's cube matrix, sequence scores and supernumeraries himself.

What series, elliptic integral and some number theory problems, he wrote the results in two notebooks.

The instability of life can't reduce his interest in mathematics. A kind neighbor old lady, look at him.

Life is difficult. I invited him home for something to eat several times during Chinese food.

According to Indian custom, his family arranged a marriage for him on 1909, and his wife was a nine-year-old child.

Girl. 19 10 years, he was 23 years old and had a family. Because he was the eldest son, he had to help his family with some expenses.

Yes, he had to try his best to find a job. Later, his friend recommended him to find an Indian official, Rao.

Rao himself is a rich Indian official and one of the founders of the Indian Mathematical Society. he thinks

Ramanujiang is not suitable for other jobs, and it is difficult to introduce jobs to Chang, so I would rather give him some every month.

Money is enough for him to live without going to work, but he can do his own research. He appreciates the mathematics of La Manu River.

Genius.

Gemanujiang had to accept the money and continue his research work. Every night in Madre.

Madras took a walk by the sea and chatted with friends as a rest. One day, an old friend met him.

Say to him: "People praise you for your mathematical genius!" " Ramanujiang laughed: "Genius? ! please

Look at my elbow! The skin on his elbow looks dark and thick. He explained that he was counting on the slate day and night.

Well, it takes too long to wipe off the words on the slate with a rag. He wiped the slate directly with his elbow every few minutes.

Words. A friend asked him why he didn't use paper to write since he had to do so many calculations. Ramanujiang said he even ate it.

It's all problems. Have money to buy a lot of paper. It turned out that Manu Jiang felt that he had to rely on others to survive.

Li was ashamed and didn't take money for a month.

Fortunately, La Manu Jiang won a scholarship. Since May of 19 13, he has been earning in 70 yuan every month.

Five rupees. Soon, his friend helped him write a letter in English to the mathematics department of the famous Cambridge University in England.

In this letter, Professor G.H.Hardy listed 120 prescriptions he had studied before.

Rational sum formula.

Professor Hardy saw some of his achievements, some of which were the results of rediscovering great mathematicians a hundred years ago.

Some are mistakes, and some are very profound and difficult. After many twists and turns, the Lamanu River finally came.

Arrive in England. Hardy thinks that to teach him modern mathematics, if he starts from scratch as usual, he is likely to pull.

Manu river talent is damaged. And he can't stay in a state of ignorance of modern mathematics. So Hardy

With his own unique method to help him study, Ramanu River finally mastered a relatively sound modern analysis theory.

Knowledge of. More than he taught Ramanu River.

Professor Ramanu He Jiang has written many important mathematical papers from 19 14 to 19 18. Because he is a

As a devout Brahmin, he is absolutely vegetarian. During his stay in England, he made it himself.

I often forget to eat my own food because of my research, and my body is getting weaker and weaker, and I often get sick later.

I have nameless pain on my face.

It was later discovered that he had an incurable disease. I lived in an English hospital for some time. Erythrocyte adsorption inhibition

The professor told a story about his illness:

One day, Hardy went to see him by taxi. The license plate number is 1729. Hadi vs Rama

Nujiang gave this figure, which seems meaningless. But Rama Jiang wanted to think and immediately answered:

"This is the smallest integer, and the sum of the cubes of two integers can be expressed in two ways. 」

( 1729= 13+ 123=93+ 103)

La Manu River is known as a mathematical prophet. It has been 54 years since he died, but he is one of them.

The results of these predictions are still being proved by mathematicians.

1920 died in Matlas on April 26th, and university of madras later established institutions of higher learning.

The Institute of Mathematics is named after him. And at 1974, prepare to serve him in front of the institute.

Dali's bust stands.

If he is wise, maybe he will say, "Don't stand up for me, please those who are hungry."

Many of the dead children will be the future Ramanu River! 」

Gauss, a great German mathematician, physicist and astronomer, is known as the "prince of mathematics".

Scientist.

C.F.Gauss (1777- 1855) is the greatest German mathematician.

The greatest and most outstanding scientist, if only based on his mathematical achievements, rarely makes achievements in a subject.

Some of his research results are not used in the branch of mathematics.

Born in a poor family.

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.

Brilliant ancient Babylonian culture

Originated in the Tigris and Euphrates rivers in Turkey.

These rivers flow southeast into the Persian Gulf. The river runs through what is now Suri.

Asia and Iraq.

The "weekly system" we live in now originated from Babylon. Babylon

People divide a year into twelve months, and seven days make up a week. At the end of the week.

One day's reduced work is devoted to religious worship, which is called the Sabbath-this is us.

It's Sunday.

We are now 24 hours, 60 minutes an hour and 60 minutes a minute.

The time division of seconds was established by the Babylonians. Divide a circle into 360 mathematically.

Ten degrees, once 60 minutes, this 66 is also Babylon's.

The contribution of.

The ancient Babylonian writing tools were very strange and were used everywhere.

Sticky mud, made into rectangular pancakes, this is their paper. And then use one end

The sharpened metal bar is written into cuneiform characters when the pen is used, forming mud.

Write on the blackboard.

Greek travelers recorded the canals built by Babylon for agricultural needs.

The grandeur of this project is amazing. And the beauty of urban architecture, frequent commercial trade.

Many people are engaged in law, religion, science, art, architecture, education and machinery.

Engineering research, which was rare in other countries at that time.

However, Babylon flourished for a while and then declined. Many cities are buried in the Yellow River.

Earth, sand and Babylon became the land of legends and myths, and people could not find them on the ground.

The trace of this country used to be a famous "hanging garden" buried in yellow mud for tens of meters.

Under the soil, there are only wild sheep running on the wasteland.

1In the 1940s, French and British archaeologists excavated this ancient city.

And get a lot of cultural relics, the world can once again witness this ancient country that disappeared on the ground.

Understand its cultural prosperity. Especially the Englishman Roald in Nineveh.

Nineveh dug into the Royal Library, and there were more than 26,000 pieces of mud in two rooms.

Blackboard books contain records of history, literature, diplomacy, business, science and medicine. Eager hope/persistence

Babylonians knew 500 kinds of drugs and how to treat diseases such as earache and ophthalmia, but biologists remembered.

Contains the names and attributes of hundreds of plants. Chemists know the characteristics of some minerals,

In addition to medicine, refined metals are also used, and the level of pottery and glass is also very good.

Tall man.

A nation with such a high level of education should be good at math, right? this

Let's talk about their contribution in this respect.

Babylonian symbol

Babylonians used two rounding methods: one was decimal and the other was sexagesimal.

Carry.

Decimalization is a method we use in our daily life, and it is an abacus.

"Everything is unified" is based on this principle.

Babylonians did not have an abacus, but invented such a "calculation tool" association.

Help calculation (Figure 1). Dig three long small grooves in the ground, or have three small ones specially.

Rotten mud, use some metal balls to represent numbers.

For example, farmers in southern Babylon gave 429 bags of wheat to the king.

Tax, farmers in the east of the city paid 253 bags of wheat. So the king's warehouse increased.

429+253 = 682 bags of grain. We can get the answer in an instant with a pen, but

It belongs to Babylon, but it was first put in small grooves on the clay board: four, two,

Nine metal balls represent 429. Then put four metal balls in a small groove.

Add 2 balls to the top, 5 balls to the middle slot, and the last slot.

Three balls.

Now there are 12 balls in the last slot, and the Babylonians will take ten.

One, add 1 ball to the middle slot-that's "one in every ten".

Finally, the number 682 on the clay tablet is the result of addition. Isn't this fun?

(Figure 2) We can teach children about the addition of large numbers in objects in this way.

Law.

At present, hexadecimal is rarely used unless we say: one hour.

When = 60 minutes, 1 minute = 60 seconds, we use decimal system in other occasions.

But you know what? It was set by the Babylonians in 360.

Five days, twelve months, twenty-nine or thirty days in a month, once every seven days.

A week, a circle has 360 degrees, an hour has 60 minutes, and a minute has 60 seconds.

Wait, let's continue to adopt it in modern times.

Archaeologists in a piece of 3 1/8 inches long, 2 inches wide and 3/4 inches thick.

Babylonian symbols were found on an inch-thick clay tablet.

There are similar symbols (Figure 4) in the middle of this clay tablet from top to bottom: readers can read.

This means: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,1,12, 13.

The blackboard is eroded by salt and dust, but you can see the right side of the blackboard.

The shapes of the first five lines are as follows:

Obviously, this should represent 10, 20, 30, 40, 50.

But then came this symbol:

If the symbols we knew before were written as:

11101.20 (missing three) 2 2 10

What does this mean? Archaeologists speculate that those symbols were shot above 10, 20, 30,

The order of 40 and 50 should be 60, 70, 80, (missing 90, 100,10), 120, 130.

Can the symbol of 1 also represent 60? If so, then 1, 10.

It means 60+ 10 = 70. And 1, 20 stands for 60+20 = 80. and

Will represent 2 × 60 = 120. Obviously, 2 10 is120+10 =130.

Such a guess is reasonable, because the Babylonians did not have the symbol of zero, and

They use hexadecimal, so the same symbol can represent 1 or 60.

There is no zero, which is easy to cause misunderstanding when counting, for example, there is.

Take 1, 20 = 1 × 60+20 = 80 or 1, 20 = 1× 602+0× 60+20 = 3620.

It was not until two thousand years ago that Babylon adopted zero.

Therefore, the image represents 2,3,0,4 1, that is, 2× 603+3× 602+41= 442841.

It can be seen that the Babylonians knew the "bit value principle" through numbers less than 60.

How to divide the Babylonians?

You can see the following correspondence from some clay tablets.

2 30 16 3,45 45 1 ,20

3 20 18 3,20 48 1 , 15

4 15 20 3 50 1 , 12

5 12 24 2,30 54 1 , 6 ,40

6 10 25 2,24

8 7,30 27 2, 13,20

9 6,40 30 2

10 6 32 1,52,30

12 5 36 1,40

15 4 40 1,30

If you dig such a blackboard writing on the land of Iraq today, you can understand what it is.

Meaning? More than forty years ago, archaeologists discovered that this was actually the Babylonian "countdown table". I

Now rewrite the above table:

You can see that this is the reciprocal of the integer n 1/n expressed by 60 points. Suppose you are 27 years old.

Corresponding to 2,13,20 means:

You will notice that all the above tables are missing: 7, 1 1, 13, 14, 17, 19, 2 1, 23, 26.

What is the reason?

So that's it: the Babylonians only listed those integers whose fractional expressions were limited in hexadecimal.

Numbers, and these integers can only be 2a3b5c (where a, b and c are integers greater than or equal to zero).

For 7, if its reciprocal is expressed by 60 digits, you will get a cyclic score, which is 8,34, 17.

8,34, 17, ... until infinity. 1 1 is the same. We get 5,27,16,2149, and repeat the above sample.

Type even infinite.

Why build such a "reciprocity table"?

We learn calculation in primary school: first we learn addition, then we learn subtraction. Learn multiplication first, then division. If you want to count now,

A ÷ b, we can turn this problem into a ×), so as long as we know the reciprocal of B, we will "

Division is sometimes faster than multiplication.

The ancient Babylonians also understood this truth, so in real life, such as irrigation and wage calculation.

If you encounter problems such as division, interest, taxation, astronomy, etc. Try to turn it into a multiplication problem to solve.

To be sure, the "countdown table" is very useful at this time.

On the Discovery of Irrational Numbers

The Pythagorean school of ancient Greece believed that any number in the world could be expressed by integer or fraction, which was their creed. One day, hippasus, a member of this school, suddenly found that the diagonal of a square with a side length of 1 was a strange number, so he studied hard and finally proved that it could not be expressed by integers or fractions. But it broke the Pythagorean creed. So Pythagoras ordered him not to reveal it. But Siberus revealed the secret. Pythagoras was furious and wanted to put him to death Siberus ran away at once, but he was caught and thrown into the sea, giving his precious life for the development of science. The numbers discovered by Siberus are called irrational numbers. The discovery of irrational numbers led to the first mathematical crisis and made great contributions to the development of mathematics.

Euclid (about 330-275 BC) was an ancient Greek mathematician. His book Elements of Geometry is famous all over the world. Euclid put the rich and varied achievements of Greek geometry accumulated since the 7th century BC into a strict and unified system. Starting from the initial definition, he listed five postulates, and deduced a series of theorems and inferences through logical reasoning, thus establishing the first axiomatic mathematical system called Euclidean geometry.

According to records, a ruler asked him if there was a simple way to learn geometry, and he replied, "In geometry, there is no avenue to pave the way for kings." This sentence later became a learning motto handed down from ancient times. Besides the Elements of Geometry, he has many other works, but most of them have been lost. The division of known numbers and circles is a preserved work.