Fengse compiled from the concave temple

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In the history of mathematics, there is a famous problem of "turning a circle into a square".

It was put forward by an ancient Greek philosopher in prison in the 5th century BC. It is about drawing a square with a compass and a scaleless ruler to make its area equal to that of the circle.

However, no one expected that this problem has been puzzling mathematicians for more than 2000 years.

In the end, it was discovered by everyone, and it was impossible to draw it.

The reason is related to π being a transcendental number-

Yes, π is a transcendental number besides the irrational number identity known to primary school students.

When it comes to transcendental number, it’s a really hard concept to understand, and no number of professional textbooks can make it easy to understand.

It can only be said that it is a class of irrational numbers, but it is more "unreasonable" than irrational numbers. .

From the 18th century to the present, back and forth for more than 200 years, a large number of mathematicians from various countries have pounced on it, looking for evidence to prove its existence and ways to judge it.

However, this "little thing" is so elusive that until now there are still many suspected figures that have not been proved and a lot of open problems related to it have not been solved.

"Terrible" is that many mathematicians said:

In fact, I know that it is useless to find out which number is beyond the number, but I just can’t stop!

Therefore, today’s article is a dry issue of mathematics, which comes from the column of Quanta Magazine (author David S. Richeson). It is the most clear and easy-to-understand article that tells the ins and outs of transcendental numbers, and is hereby compiled.

Then, blessed are those who like mathematics today-

Love beyond a few times again, and see how it was discovered by mathematicians and how it was related to that Millennium problem.

The Unsolvable Problem of "Turning a Circle into a Square"

Mathematician Leopold Kronecker famously said:

God himself created natural numbers, and the rest came from human power.

Nowadays, although all kinds of numbers have a fascinating and complicated history, basically everyone has a clear definition of them.

For example, integers are positive integers and zero plus negative integers. Rational number is a number that can be expressed as the ratio of two integers, including integers, finite decimals and infinite cyclic decimals.

If the decimal places of this ratio are never divided and never repeated, it is an irrational number.

Then rational numbers and irrational numbers together form real numbers, and real numbers and imaginary numbers form complex numbers.

Among them, today, we all agree that rational numbers were discovered by hippasus, who lived around the 5th century BC (and he died because of it).

But in fact, his discovery belongs to geometric meaning rather than arithmetic.

He proved that you can find two line segments that cannot be divided into several equal lengths, such as the sides and diagonals of a square.

Today we call this incommensurability, which means that their lengths are not rational multiples of each other.

Therefore, the diagonal length of a square is √2 times the side length, and this √2 is called an irrational number.

Undoubtedly, using compasses and straightedge, we can draw a line segment with any positive rational length.

However, some unreasonable lengths are also acceptable. For example, in the famous golden ratio (1+√5)/2, draw a regular Pentagon with a side length of 1 and take its diagonal.

In 1637, the famous mathematician Descartes proved:

As long as a number can be represented by integer operations of addition, subtraction, multiplication, division and square root, then we can draw the corresponding line segments (such as all positive rational numbers, as well as the √2 and the golden ratio just mentioned).

Therefore, if we can express π with an exact formula, can we solve the problem of "turning a circle into a square"!

The problem is, I can’t find it.

In a blink of an eye, 200 years have passed (two centuries have passed).

A French mathematician named Pierre Wantzel finally proved:

If a number can be drawn by line segments, that is, it can be constructed, then it must also be the root of a polynomial that cannot be further factorized or simplified, and the degree of the highest term of this polynomial is a power of 2 (such as 2, 4, 8, 16, etc.).

For example:

√2 is the root of the polynomial x2–2, and the golden ratio is the root of x2–x–1, so they can all be drawn.

And like 3; √2 is the root of the polynomial x3–2, which does not meet the condition of the second power, so it is impossible to draw a line segment of this length.

However, even if we know this law, we still haven’t found the exact expression of π, and the problem of turning a circle into a square is still unresolved.

It was finally proved that it could not be drawn at all.

In fact, the key to solving the problem of turning a circle into a square is just like mathematicians divided real numbers into rational numbers and irrational numbers before-complex numbers need to be divided into two sets.

For complex numbers, many of them are equal to the roots of polynomials with integral coefficients, which mathematicians call algebraic numbers.

Every rational number is an algebraic number, and so are some irrational numbers, such as 3; √2, and even the imaginary number I, it counts, because it is the root of x2+1.

Not all complex numbers are like this, so the values that do not belong to this category are called transcendental numbers.

However, it is not obvious whether a transcendental number exists, and it is also very challenging to prove whether a given number is transcendental, because it is necessary to disprove that a number is not the root of any integer coefficient polynomial.

The first breakthrough point was born in 1844, and a French mathematician named joseph liouville thought of such an indirect method:

Since irrational numbers can’t be well approximated by rational numbers, if I find a number that can be infinitely approximated by fractions with smaller denominators, it must be something else: transcendental numbers.

So, Joseph Liouville constructed such a number:

L=0.1100010000000000000000010…

It has only 0 and 1, where the position of 1 appears in turn by "n!" Decide that the position of the first 1 is equal to "1!" , that is, 1, and the second is equal to "2!" , appears in the second place, and the third is equal to "3!" , in the sixth place, and so on.

This ingenious design makes the scores of 1/10, 11/100 and 110001/1000000 very close to the value of L itself.

This proves the existence of transcendental numbers (that is, he created this L).

However, the number π does not satisfy Joseph Liouville’s condition (it cannot be infinitely approximated by rational numbers), so it is not clear whether π is beyond the number.

The important breakthrough came in 1873, nearly 30 years later.

A French mathematician named Charles Hermite designed a clever method to prove that the base e of natural logarithm is transcendental.

This is the first transcendental number that is not artificially designed, which makes Lin Deman, a German mathematician, finally prove that π is a transcendental number on the basis of his theory nine years later.

First, he proved that as long as D is a nonzero algebraic number, ed is a transcendental number.

In other words, if ed is an algebraic number, but d is either 0 or transcendental.

In this case, based on Euler’s identity, which is recognized as the most perfect formula in mathematics, eπ i = 1, the category of π can be deduced.

Because -1 is an algebraic number, πi must be a transcendental number based on the theorem just now, and π must be transcendental because I is also an algebraic number.

In this way, "big things are not good"-

π is a transcendental number, which means π does not meet the definition of "line segment that can be drawn" mentioned in the previous paragraph, so the problem of turning a circle into a square put forward by ancient Greek philosophers thousands of years ago is impossible to realize.

The story is not over yet.

The story of turning a circle into a square is over, but the exploration of transcendental numbers has just begun.

Shortly after E was proved to be a transcendental number, another mathematician proved that infinite numbers actually have different sizes, but the infinity of rational numbers is the same as the infinity of integers. Such a set is called "countably infinite".

However, the set of real numbers and irrational numbers is larger, which is uncountable infinity; At the same time, although the algebraic number set contains all rational numbers and infinitely many irrational numbers, it is still an infinitely small and countable infinite set.

Therefore, its complement, that is, transcendental number, is uncountable and infinite.

In other words, most real numbers and complex numbers are transcendental numbers.

-Although there are so many excesses, by the turn of the 20th century, mathematicians could only identify a few of them.

In 1900, the famous German mathematician David Hilbert listed a famous list of 23 most important unsolved mathematical problems, the seventh of which was:

It is proved that when A is an algebraic number and is not equal to 0 or 1, and B is an algebraic irrational number, ab is a transcendental number.

In 1929, Aleksandr Gelfond, a young Russian mathematician, proved that B = I √ r and r are special cases of positive rational numbers, which means that eπ is a transcendental number.

This result is surprising, because according to the theorem, e and π are not algebraic numbers.

However, by manipulating Euler’s identity skillfully again, we know what’s going on:

Soon after, another mathematician discovered that 2√2 was a transcendental number. Then, the seventh problem listed by David Hilbert was solved by two mathematicians independently.

Later, Alan Baker, a mathematician, published a series of articles, summarizing the achievements of these people, which made us have a deeper understanding of transcendental numbers, and therefore won the Fields Prize in 1970.

Of course, at the age of 31, his greatest achievement in winning this honor is to prove that two values such as the following are also transcendental numbers:

Until now, there are many open questions about transcendental numbers, and there are many concrete numbers that seem to be "very transcendental", which have never been proved, such as "eπ", "e+π", "ee", "π π" and "π e".

But as said at the beginning, the mathematician said:

It may not be useful to know who is above the number, but if we have the ability to know but choose not to know, it is definitely unacceptable.

Seriously respect.

Original link:

How Math Achieved Transcendence

End—

Advances in science and technology meet every day ~

Original title: "The irrational number in this irrational number makes mathematicians call" it can’t stop at all ""

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Recently, in Henan Satellite TV’s "Walking in Henan Xianglong Eighteen Bowls" program, the first women’s iron flower team appeared, and the 1600-degree molten iron was turned into a colorful sky, which was praised by netizens. Among them, there are cultural bloggers from the media and members of the fire brigade, as well as the lead singer of the band and the head of the Henan Opera Troupe.

Wu Qiao, a member of the women’s Tiehua team and a member of the Henan Zhumadian Fire Rescue Detachment, told the Beijing Youth Daily reporter that during the official performance, she changed from nervous to excited, and the more confident she became, the more she wanted to fight. She feels that she seems to have become different after playing iron flower, and the strength she once lost seems to be coming back. Ma Dandan, another lead singer of Blu-ray Band, said that anyone who says that women are not as good as men can play beautiful iron flowers. "No matter how hard and tired you are when you do it, the moment when the dazzling iron flower is finally hit, I feel that all the efforts are worthwhile."

The appearance of the first women’s tiehua team attracted attention from bloggers, fire workers, the head of Henan Opera and the lead singer of the band.

As high as 1600 degrees, molten iron was hit into the air by willow sticks, turning it into colorful flowers all over the sky, landing to produce gold, and striking iron flowers is a traditional folk fireworks spread in Henan and Jin regions, and it is also a national intangible cultural heritage. Previously, since September, the cultural media bloggers had expressed their best wishes of "exorcising evil spirits and praying for good fortune" in a short video, learning from Yang Jianjun, the national representative inheritor of Queshan Tiehua, and became the first female inheritor of Queshan Tiehua.

Before the Spring Festival, Henan Satellite TV decided to form a women’s iron flower team and let them perform iron flowers in the program "Walking 18 bowls of Henan Xianglong". September became the first member of this team, and Yang Jianjun became the instructor.

Wu Qiao, a member of the fire rescue detachment in Zhumadian, Henan Province, is mainly responsible for propaganda work in the team. In order to experience the bravery of firefighters who rushed into the fire, she joined the women’s iron flower team. "No matter what difficulties you encounter, you will always believe that you can overcome everything. The more difficult it is, the more forward you will be."

"These women are not as good as boys." As the Mulan in Henan Opera sings, men can strike while the iron is hot, and so can women. It’s never harder than usual to play broadsword, is it? With this in mind, Fan Shengnan, head of Anyang Youth Henan Opera Troupe, also joined the women’s iron flower team.

Another member of the women’s iron flower team is Ma Dandan, the lead singer of Blu-ray Band. She told the reporter of Beiqing Daily that she joined the women’s iron flower team to participate in the performance. At first, she took advantage of her family and told her parents when she arrived in Zhumadian. After her parents knew about it, they were worried and questioned about her, thinking that her small body could not beat the iron flower. "If they say so, I will try, I must type and try my best."

During the training, two willow sticks experienced thousands of knocks, from hitting stones to sand to drawing water, and finally the water was replaced by molten iron. Four members of the women’s team were finally ready to face the molten iron. In late January, when the program was recorded, they beat the molten iron at 1600 degrees into a colorful sky under the willow flower shed, accompanied by a festive dragon dance, and their wonderful performance attracted many viewers. On February 7 th, the program was officially broadcast, and the women’s iron flower team also won praise from netizens.

When training, I hold a willow stick weighing more than ten kilograms and use stones, sand and water instead of 1600 degrees of molten iron.

Before the training, Ma Dandan watched the video of Yang Jianjun playing iron flowers at home, and did strength training in the gym in advance to enhance his arm strength. Fan Shengnan, on the other hand, took time out to practice after work, holding flowerpots and water at home to exercise his arm strength. "The iron flower requires that the arm must be stable, and the running of the water can’t be spilled. I will practice the basic skills of the catwalk in Henan Opera while practicing the water."

After the training, the players held willow sticks and beat them many times a day. Before training, it is extremely dangerous for players to strike molten iron. During training, players will first replace molten iron with stones, sand and water. Hit the stone into the air more than ten meters high with a willow stick to train the strength of hitting. Beating sand can simulate the feeling of molten iron sprinkled on the body and train the accuracy of hitting. And water is the closest to molten iron, which is very challenging. It is necessary to produce the effect of "drizzle, such as mist dispersing". To reach such a level, the beating of willow sticks requires a delicate combination of strength and angle, and nothing else is better than being skilled.

When choosing willow sticks, Wu Qiao picked the biggest pair, weighing more than ten kilograms. After training every morning, at noon, she even had difficulty holding chopsticks because her hands were shaking badly. "I think it’s more difficult when training, but it’s not so difficult when performing formally."

Wu Qiao told the reporter of Beiqing Daily that during the training, they didn’t have protective gear on them. When hitting stones, they had to avoid the fallen stones themselves. When hitting sand, they often got sand in their hair and clothes. In order to make the usual training more interesting, they will cheer each other up and compete, holding the willow sticks longer than anyone else and hitting the stones and sand higher than anyone else.

During the training, it was the cold winter, and the players were subjected to the double tests of cold environment and physical strength. Ma Dandan’s strength is relatively small, and her teammates call her "Dan Dali". The other players looked relaxed in training, but Ma Dandan felt hard, but she said she wanted to give up and kept training. "I think some things still need to be adhered to."

As a veteran, Wu Qiao’s strength training has a certain foundation, but she still feels that hitting willow sticks every day has increased her arm strength a lot. Her child is less than one year old. When she comes home from training every day, she feels that the child seems to be "lighter" when she holds the child.

Getting applause from the audience during the formal performance changed from nervousness to excitement, and the more you play, the more confident you are and the more you want to play.

On the evening of January 25th, the women’s Tie Hua Team came to Zhumadian Royal Station Scenic Area to perform Tie Hua, which was also officially recorded. In order to ensure the safety of the performance, the scene was equipped with fire engines, ambulances, and public security on duty. In order to make the performance better, the street lamps nearby were temporarily extinguished.

Before playing, Yang Jianjun asked, "Are you afraid of playing iron flowers?" The female players answered loudly, "Not afraid!" After getting ready, the female players went on stage one by one. They wore their hair up, gourd ladle caps, cowhide fire-proof shoes, linen performance clothes and goggles. Under the willow branch flower shed with red cloth strips for praying, molten iron was shot into the air, fell on the flower shed and bounced off, scattering sparks all over the sky, accompanied by dragon dance performances, full of flavor.

Wu Qiao told the reporter of Beiqing Daily that during the official performance, the flower stick they took, that is, the willow stick, was smaller than that during training. She held it easily, but she was still a little nervous. Everyone takes turns to play, and the next person will play before the previous one has finished playing. "I don’t want to back down, and I can’t back down. I try to adjust my state with every stick. Every time I finish receiving hot metal, I tell myself ‘ Forward, forward, forward ’ , and then rush. "

When Ma Dandan first started playing, one of the iron flowers missed the flower stand, and she was in a hurry to cry. With everyone’s comfort, she quickly adjusted her state, playing better and better, and was congratulated and encouraged by Yang Jianjun. When the program was broadcast, she looked at the iron flower she played in the video and felt that all the training was worth it.

During the performance, their teacher Yang Jianjun also encouraged them. "The more I played, the more confident I became. The more I played, the more I wanted to play. The more I played, the more excited I became. I couldn’t stop … … Until the hot metal is used up, everyone probably hit about 20 sticks. " Wu Qiao said that at that time, many audiences applauded and she was deeply infected. "I kept waving a flower stick and screaming, and I felt very powerful."

Fan Shengnan said that at that time, in that atmosphere, the players were basically excited and cried. "There is no way to describe the feelings at that time, and the emotions are surging. I feel that the whole body and mind are shaken into it, and I don’t feel nervous."

Some players were burned out of their clothes while performing. "The dazzling iron flower was beaten out and it felt that all the efforts were worth it."

During the formal performance, the iron flower fell on Fan Shengnan’s body, and the clothes were scalded out of the hole. Others asked if it hurt. She smiled and said, "It’s not so melodramatic, it doesn’t matter if you get burned, it’s worth it."

It was only after the performance that Wu Qiao discovered that two fingernail-sized holes were burnt out on her clothes by iron flowers, and two small scars were left on her skin. "There was no bleeding, and I didn’t feel it during the performance. It was just the feeling of stabbing, which was not very painful. Later, I touched the hole and found that the thermal underwear inside had been burned by hot metal. During the performance, there were also iron flowers splashing into my mouth. I felt that I was bounced off when I was collapsed, and I didn’t get burned. "

It is dangerous to hit the iron flower. If the iron flower is not scattered enough, it will burn on the skin. However, after solid training, it will have a drizzle effect, and even if it falls on the body, the burn will not be serious. According to Wu Qiao, it’s inevitable that you will get a sesame-sized burn when you are playing iron flowers. She also noticed that some senior brothers were shirtless while performing, and there were some scars left by burns on their bodies.

Finally, it is not easy for women to play iron flowers compared with men. Wu Qiao thinks that women need more courage and perseverance to play iron flowers and practice harder. Ma Dandan feels that as long as she is brave enough, dare to pick up hot metal and dare to fight, women can also play gorgeous iron flowers. "Who said that women are not as good as men? I think everyone can play beautiful iron flowers. No matter how hard and tired you are when you do it, the moment when the dazzling iron flower is finally hit, I feel that all the efforts are worthwhile. "

Wu Qiao hopes to pray for his family during the Spring Festival, and all his family members will be healthy in the new year. "By participating in this performance, I feel as if I have become different. The strength I once lost seems to have come back, and I seem to have found myself again."

Ma Dandan said that as a native of Henan, she felt honored to be able to participate in the inheritance of the intangible heritage. As a musician, she hopes to create more good music works in the new year, and also hopes that fans and friends will be happy and happy in the new year. "I have personally experienced such a dangerous thing and such a shocking picture, and I feel very proud. I have drilled such a dangerous flower shed, and no matter what difficulties I encounter in the future, it is nothing. "

Intern Jin Jing

Text/Beijing Youth Daily reporter Dai Youqing

1905 movie network news On March 31st, the Ministry of Education officially announced the schedule of the national college entrance examination in 2020. "With the consent of the CPC Central Committee and the State Council, the unified national college entrance examination in 2020 will be postponed for one month, and the examination time will be from July 7th to 8th."

　

This is the first time that the college entrance examination in China has been postponed for one month since it was resumed in 1977. Many candidates ridiculed, "I didn’t expect to sleep, and the time was lengthened."Undoubtedly, the college entrance examination has influenced and changed the fate of several generations of Chinese, and it has already become a major event that affects the "heart" of thousands of families in China.

　

Everyone still remembers that the hit drama "Little Joy" in 2019 took the "College Entrance Examination" as the incision, and presented the group images of China families facing the crucial year of their children’s growth to the audience. In this family-based sprint towards the most important exam in life, parents and children are also experiencing countless "quizzes" from life.No one will forget that at the finale, the parents of the three protagonists all appeared outside the examination room with great solemnity to cheer for their children.

　

Each of us has experienced those days, and when we look back, we can always feel the same way. That struggling summer was a time for everyone to make a final summary of their youth.

Of course, in addition to "Little Joy", many film and television works involve the plot of the college entrance examination, and even the plot in many films is exclaimed by the audience. "I feel that my youth has been peeped by the director."

Little by little in the third year of high school

When it comes to senior three, what do you think of?Maybe it’s the "Five-year College Entrance Examination Three-year Simulation" that hasn’t been finished until graduation, or the exam paper that can’t be finished every day? It seems that everyone has his own answer, butIn the classroom, everyone is familiar with the scene of in-class testing from time to time.

　

Everyone wants to pass the test questions again and again, so that they can get better results in actual combat. The short film "I want to enter the top ten" is in line with everyone’s mentality. Although the male master ranks first from the bottom in the exam, he is still in a good state in the elite class of the grade. However, when small fish are in a big fish tank, they always hope to become stronger.

　

Similarly, there are women who study hard for the university they want to go to, although their past achievements are not brilliant. Looking back now, that time was the most wonderful, and I realized that I could do it for one thing without hesitation.

　

Or maybe, like that, three or two friends agreed with each other to work hard for the sake of going to the same university in the future. A small agreement has become my biggest motivation.

　

Whether for yourself or with your friends, all the vows become particularly important at this time. The swearing-in meeting of senior three is a ceremony that will not fall behind. As in the school, student representatives lead everyone to swear together, and everything they are trying to pay now is for a better future.

　

On the day of the exam, all the trips are the same as in the school. Everyone can only enter the venue with the necessary products for the exam, sit in their own seats, and answer questions seriously, calmly and carefully.

　

When the exam is over, it will usher in the craziest time for everyone. Like in Young Style, at this moment, all the papers and homework can finally be thrown away and vented. The "snow of papers" floating between the teaching buildings may be the most beautiful scene in June.

　

After the exam, we discovered that the hardest time was waiting for the release of the list. It’s just that when the list is released, the agreement with friends at that time will always fail for various reasons. Everyone will meet at everyone’s "secret base" like Li, and have the most extravagant farewell.

Here, we bid farewell to high school and youth, and welcome the university that will create more miracles for four years.

The college entrance examination is not a battle for one person.

When it comes to the college entrance examination, we will not forget our closest teachers except those thick science and education books and examination papers one after another.

　

In Young Style, Mr. Sa, played by Amanda, was completely recreated as a real senior three teacher.

　

Listening to the slogans from her mouth, such as "open your eyes and start to compete", "carry it if you can, and carry it if you can’t" and "if you raise one point, you may kill tens of thousands of people", I was immediately pulled back to the classroom of senior three. That kind of tension against time arises spontaneously.

　

Even at the parents’ meeting, she said, "If you want a divorce, you can make do with it, and then leave after the exam." It seems absurd, but this line was originally from a documentary.

　

Just like her, there are Miao teachers played by Zhong and Yu Qian.

　

Like all senior three class teachers, he can’t wait to stare at the students all the time and don’t allow them to relax at all. But under the harsh appearance of the teacher is a warm heart. He will give lessons to students who have no money to tutor, and does not discriminate against those "bad children" who are avoided by everyone. He makes all students have dreams.

　

Teacher Sa and Teacher Miao are the most common examples around us in the past. At school, everyone may scold them behind their backs and blame them for making a fuss about all kinds of things, but after graduation, they gradually realize their painstaking efforts.

　

Of course, there are some teachers who jump out of the traditional education mode. Although the film is based on the college entrance examination, it rarely mentions the pain of the college entrance examination. Fan Liben, played by Fang Zhongxin, solved the problem of students’ own growth with his own unique education method when students were ready to give up the college entrance examination.

　

In addition to teachers, parents are also more nervous than candidates, and regard senior three as the last battlefield. The spirit of "Meng Mu’s Three Movements" is all reflected in them. In Young Style, Yongmei plays the incredibly mother, who moved her home to the vicinity of the school in order to make her children have a better environment for going to and from school.

　

In "Little Joy", Christina plays the mother who moves for the sake of her two children. However, most parents are closer to the mother played by Tao Hong. In order to make their children’s nutrition keep up with the physical energy consumed by staying up late to review, they try their best to prepare the best food for their children.

　

During the preparation period, the candidates’ emotions are very easily influenced by their surrounding families. In the movie, the man has been avoiding the college entrance examination and is depressed. His uncle told him, "Why do people like to immerse themselves in unrealistic fantasies? Because there are too many problems and troubles in real life, many people don’t think about solving problems, but choose to escape, because escaping is the simplest and actually the most useless. "

　

He slowly found the meaning of the college entrance examination while getting along with his great uncle, and gradually realized that he still had dreams to pursue. At the end of the film, we didn’t tell the audience the results of the male college entrance examination, but when the college entrance examination is over, we will understand that the most precious thing is always preparing for the college entrance examination.

We will all go through a college entrance examination. I hope everyone will not forget it and will not leave regrets. This time once seemed difficult, but it was actually the happiest. For the students who are about to face the college entrance examination in 2020, Xiao Dianjun also wishes everyone in advance, calmly taking the exam and getting the best results.