Pi is a constant. What is pi and what is its history

(), and it became generally accepted after the work of Euler. This designation comes from the initial letter of the Greek words περιφέρεια - circle, periphery and περίμετρος - perimeter.

Ratings

• 510 signs after aim: π ≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 306 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606 606AR 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 964 428 810 975 665 933 446 128 475 648 233 786 783 165 271 201 909 145 648 566 923 460 213 393 607 260 249 141 273 724 587 006 606 315 588 174 881 520 920 962 829 254 091 715 364 367 892 590 360 011 330 548 820 466 521 384 146 951 941 511 609 433 057 270 365 759 591 9 381 932 611 793 105 118 548 074 462 379 962 749 567 351 885 752 724 891 227 938 183 011 949 129 833 673 362…

Properties

Ratios

There are many formulas with the number π:

• Wallis formula:

• Euler's identity:

• T. n. "Poisson integral" or "Gauss integral"

Transcendence and irrationality

Unresolved issues

• It is not known whether the numbers π and e algebraically independent.
• It is not known whether the numbers π + e , π − e , π e , π / e , π e , π π , e e transcendent.
• Until now, nothing is known about the normality of the number π; it is not even known which of the digits 0-9 occur in the decimal representation of the number π an infinite number of times.

Calculation history

and Chudnovsky

Mnemonic rules

In order not to make mistakes, We must read correctly: Three, fourteen, fifteen, Ninety-two and six. You just have to try And remember everything as it is: Three, fourteen, fifteen, Ninety-two and six. Three, fourteen, fifteen, nine, two, six, five, three, five. To engage in science, Everyone should know this. You can just try and repeat more often: "Three, fourteen, fifteen, Nine, twenty-six and five."

2. Count the number of letters in each word in the phrases below ( ignoring punctuation marks) and write down these numbers in a row - not forgetting the decimal point after the first digit "3", of course. Get an approximate number of Pi.

This I know and remember perfectly: And many signs are superfluous to me, in vain.

Who, jokingly, and soon wishes Pi to know the number - already knows!

So Misha and Anyuta ran to Pi to find out the number they wanted.

(The second mnemonic is correct (with rounding of the last digit) only when using pre-reform orthography: when counting the number of letters in words, hard signs must be taken into account!)

Another version of this mnemonic notation:

This I know and remember very well:
Pi many signs are superfluous to me, in vain.
Let's trust the vast knowledge
Those who have counted, numbers armada.

If you follow the poetic size, you can quickly remember:

Three, fourteen, fifteen, nine two, six five, three five
Eight nine, seven and nine, three two, three eight, forty six
Two six four, three three eight, three two seven nine, five zero two
Eight eight and four nineteen seven one

funny facts

Notes

See what "Pi" is in other dictionaries:

number- Reception Source: GOST 111 90: Sheet glass. Specifications original document See also related terms: 109. Number of betatron oscillations ... Dictionary-reference book of terms of normative and technical documentation

Ex., s., use. very often Morphology: (no) what? numbers for what? number, (see) what? number than? number about what? about the number; pl. what? numbers, (no) what? numbers for what? numbers, (see) what? numbers than? numbers about what? about mathematics numbers 1. Number ... ... Dictionary of Dmitriev

NUMBER, numbers, pl. numbers, numbers, numbers, cf. 1. A concept that serves as an expression of quantity, something with the help of which objects and phenomena are counted (mat.). Integer. Fractional number. named number. Prime number. (see simple1 in 1 value).… … Explanatory Dictionary of Ushakov

An abstract designation, devoid of special content, of any member of a certain series, in which this member is preceded or followed by some other definite member; an abstract individual feature that distinguishes one set from ... ... Philosophical Encyclopedia

Number- Number is a grammatical category that expresses the quantitative characteristics of objects of thought. The grammatical number is one of the manifestations of a more general linguistic category of quantity (see the Linguistic Category) along with a lexical manifestation (“lexical ... ... Linguistic Encyclopedic Dictionary

A number approximately equal to 2.718, which is often found in mathematics and science. For example, during the decay of a radioactive substance after time t, a fraction equal to e kt remains from the initial amount of substance, where k is a number, ... ... Collier Encyclopedia

BUT; pl. numbers, villages, slam; cf. 1. A unit of account expressing one or another quantity. Fractional, integer, simple hours. Even, odd hours. Count as round numbers (approximately, counting as whole units or tens). Natural hours (positive integer ... encyclopedic Dictionary

Wed quantity, count, to the question: how much? and the very sign expressing quantity, the figure. Without number; no number, no count, many many. Put the appliances according to the number of guests. Roman, Arabic or church numbers. Integer, contra. fraction. ... ... Dahl's Explanatory Dictionary

Mathematicians all over the world eat a piece of cake every year on March 14 - after all, this is the day of Pi, the most famous irrational number. This date is directly related to the number whose first digits are 3.14. Pi is the ratio of the circumference of a circle to its diameter. Since it is irrational, it is impossible to write it as a fraction. This is an infinitely long number. It was discovered thousands of years ago and has been constantly studied ever since, but does Pi have any secrets left? From ancient origins to an uncertain future, here are some of the most interesting facts about pi.

Memorizing Pi

The record for remembering numbers after the decimal point belongs to Rajveer Meena from India, who managed to remember 70,000 digits - he set the record on March 21, 2015. Before that, the record holder was Chao Lu from China, who managed to memorize 67,890 digits - this record was set in 2005. The unofficial record holder is Akira Haraguchi, who videotaped his repetition of 100,000 digits in 2005 and recently posted a video where he manages to remember 117,000 digits. An official record would only become if this video was recorded in the presence of a representative of the Guinness Book of Records, and without confirmation it remains only an impressive fact, but is not considered an achievement. Mathematics enthusiasts love to memorize the number Pi. Many people use various mnemonic techniques, such as poetry, where the number of letters in each word is the same as pi. Each language has its own variants of such phrases, which help to remember both the first few digits and a whole hundred.

There is a Pi language

Fascinated by literature, mathematicians invented a dialect in which the number of letters in all words corresponds to the digits of Pi in exact order. Writer Mike Keith even wrote a book, Not a Wake, which is completely written in the Pi language. Enthusiasts of such creativity write their works in full accordance with the number of letters and the meaning of the numbers. This has no practical application, but is a fairly common and well-known phenomenon in the circles of enthusiastic scientists.

Exponential Growth

Pi is an infinite number, so people, by definition, will never be able to figure out the exact numbers of this number. However, the number of digits after the decimal point has increased greatly since the first use of the Pi. Even the Babylonians used it, but a fraction of three and one eighth was enough for them. The Chinese and the creators of the Old Testament were completely limited to the three. By 1665, Sir Isaac Newton had calculated 16 digits of pi. By 1719, French mathematician Tom Fante de Lagny had calculated 127 digits. The advent of computers has radically improved man's knowledge of Pi. From 1949 to 1967, the number of digits known to man skyrocketed from 2037 to 500,000. Not so long ago, Peter Trueb, a scientist from Switzerland, was able to calculate 2.24 trillion digits of Pi! This took 105 days. Of course, this is not the limit. It is likely that with the development of technology it will be possible to establish an even more accurate figure - since Pi is infinite, there is simply no limit to accuracy, and only the technical features of computer technology can limit it.

Calculating Pi by hand

If you want to find the number yourself, you can use the old-fashioned technique - you will need a ruler, a jar and string, you can also use a protractor and a pencil. The downside to using a jar is that it has to be round, and accuracy will be determined by how well the person can wrap the rope around it. It is possible to draw a circle with a protractor, but this also requires skill and precision, as an uneven circle can seriously distort your measurements. A more accurate method involves the use of geometry. Divide the circle into many segments, like pizza slices, and then calculate the length of a straight line that would turn each segment into an isosceles triangle. The sum of the sides will give an approximate number of pi. The more segments you use, the more accurate the number will be. Of course, in your calculations you will not be able to come close to the results of a computer, nevertheless, these simple experiments allow you to understand in more detail what Pi is in general and how it is used in mathematics.

Discovery of Pi

The ancient Babylonians knew about the existence of the number Pi already four thousand years ago. The Babylonian tablets calculate Pi as 3.125, and the Egyptian mathematical papyrus contains the number 3.1605. In the Bible, the number Pi is given in an obsolete length - in cubits, and the Greek mathematician Archimedes used the Pythagorean theorem to describe Pi, the geometric ratio of the length of the sides of a triangle and the area of ​​\u200b\u200bthe figures inside and outside the circles. Thus, it is safe to say that Pi is one of the most ancient mathematical concepts, although the exact name of this number has appeared relatively recently.

A new take on Pi

Even before pi was related to circles, mathematicians already had many ways to even name this number. For example, in ancient mathematics textbooks one can find a phrase in Latin, which can be roughly translated as "the quantity that shows the length when the diameter is multiplied by it." The irrational number became famous when the Swiss scientist Leonhard Euler used it in his work on trigonometry in 1737. However, the Greek symbol for pi was still not used - it only happened in a book by the lesser-known mathematician William Jones. He used it as early as 1706, but it was long neglected. Over time, scientists adopted this name, and now this is the most famous version of the name, although before it was also called the Ludolf number.

Is pi normal?

The number pi is definitely strange, but how does it obey the normal mathematical laws? Scientists have already resolved many questions related to this irrational number, but some mysteries remain. For example, it is not known how often all digits are used - the numbers from 0 to 9 should be used in equal proportion. However, statistics can be traced for the first trillion digits, but due to the fact that the number is infinite, it is impossible to prove anything for sure. There are other problems that still elude scientists. It is possible that the further development of science will help shed light on them, but at the moment this remains beyond the limits of human intelligence.

Pi sounds divine

Scientists cannot answer some questions about the number Pi, however, every year they understand its essence better. Already in the eighteenth century, the irrationality of this number was proved. In addition, it has been proved that the number is transcendental. This means that there is no definite formula that would allow you to calculate pi using rational numbers.

Dissatisfaction with Pi

Many mathematicians are simply in love with Pi, but there are those who believe that these numbers have no special significance. In addition, they claim that the number Tau, which is twice the size of Pi, is more convenient to use as an irrational one. Tau shows the relationship between the circumference and the radius, which, according to some, represents a more logical method of calculation. However, it is impossible to unambiguously determine anything in this matter, and one and the other number will always have supporters, both methods have the right to life, so this is just an interesting fact, and not a reason to think that you should not use the number Pi.

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1. The relevance of the work.

In an infinite number of numbers, as well as among the stars of the Universe, separate numbers and their whole “constellations” of amazing beauty stand out, numbers with unusual properties and a peculiar harmony inherent only to them. You just need to be able to see these numbers, notice their properties. Look closely at the natural series of numbers - and you will find in it a lot of amazing and outlandish, funny and serious, unexpected and curious. The one who looks sees. After all, even on a summer starry night, people will not notice ... radiance. The North Star, if they do not direct their gaze to a cloudless height.

Moving from class to class, I got acquainted with natural, fractional, decimal, negative, rational. This year I studied irrational. Among the irrational numbers there is a special number, the exact calculations of which have been carried out by scientists for many centuries. I met it back in the 6th grade while studying the topic “Circumference and area of ​​a circle”. Attention was focused on the fact that quite often we will meet with him in the lessons in the senior classes. Practical tasks for finding the numerical value of the number π were interesting. The number π is one of the most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. Many interesting facts are connected with the number π, so it is of interest to study.

Having heard a lot of interesting things about this number, I myself decided, by studying additional literature and searching the Internet, to find out as much information as possible about it and answer problematic questions:

How long have people known about pi?

Why is it necessary to study it?

What interesting facts are associated with it

Is it true that the value of pi is approximately 3.14

Therefore, in front of me I put goal: explore the history of the number π and the significance of the number π at the present stage of development of mathematics.

Tasks:

Study the literature in order to obtain information about the history of the number π;

Establish some facts from the "modern biography" of the number π;

Practical calculation of the approximate value of the ratio of the circumference of a circle to its diameter.

Object of study:

Object of study: The number of PI.

Subject of study: Interesting facts related to the number PI.

2. The main part. The amazing number pi.

No other number is as mysterious as "Pi" with its famous never ending number series. In many areas of mathematics and physics, scientists use this number and its laws.

Of all the numbers that are used in mathematics, in the natural sciences, in engineering, and in everyday life, few numbers receive as much attention as the number pi. One book says, “Pi is capturing the minds of scientific geniuses and amateur mathematicians all over the world” (“Fractals for the Classroom”).

It can be found in probability theory, in solving problems with complex numbers, and in other areas of mathematics that are unexpected and far from geometry. The English mathematician August de Morgan once called "pi" "... the mysterious number 3.14159... that climbs through the door, through the window and through the roof." This mysterious number, associated with one of the three classic problems of Antiquity - the construction of a square whose area is equal to the area of ​​a given circle - entails a trail of dramatic historical and curious entertaining facts.

Some even consider it one of the five most important numbers in mathematics. But, as the book Fractals for the Classroom notes, for all the importance of pi, “it is difficult to find areas in scientific calculations that require more than twenty decimal places of pi.”

3. The concept of pi

The number π is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. The number π (pronounced "pi") is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. Denoted by the letter of the Greek alphabet "pi".

Numerically, π begins as 3.141592 and has an infinite mathematical duration.

4. The history of the number "pi"

According to experts, this number was discovered by the Babylonian Magi. It was used in the construction of the famous Tower of Babel. However, insufficiently accurate calculation of the value of Pi led to the collapse of the entire project. It is possible that this mathematical constant underlay the construction of the legendary Temple of King Solomon.

The history of the number pi, which expresses the ratio of the circumference of a circle to its diameter, began in ancient Egypt. Area of ​​circle diameter d Egyptian mathematicians defined as (d-d/9) 2 (this notation is given here in modern symbols). From the above expression, we can conclude that at that time the number p was considered equal to the fraction (16/9) 2 , or 256/81 , i.e. π = 3,160...

In the holy book of Jainism (one of the oldest religions that existed in India and arose in the 6th century BC), there is an indication from which it follows that the number p at that time was taken equal, which gives a fraction 3,162... Ancient Greeks Eudoxus, Hippocrates and other measurements of the circle were reduced to the construction of a segment, and the measurement of the circle - to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples have tried to express the ratio of the circumference to the diameter of a rational number.

Archimedes in the 3rd century BC. substantiated in his short work "Measurement of the circle" three positions:

Any circle is equal in size to a right triangle, the legs of which are respectively equal to the circumference and its radius;

The areas of a circle are related to a square built on a diameter, as 11 to 14;

The ratio of any circle to its diameter is less than 3 1/7 and more 3 10/71 .

According to precise calculations Archimedes the ratio of circumference to diameter is between the numbers 3*10/71 and 3*1/7 , which means that π = 3,1419... The true meaning of this relationship 3,1415922653... In the 5th century BC. Chinese mathematician Zu Chongzhi a more accurate value of this number was found: 3,1415927...

In the first half of the XV century. observatories Ulugbek, near Samarkand, astronomer and mathematician al-Kashi calculated pi with 16 decimal places. Al-Kashi made unique calculations that were needed to compile a table of sines with a step of 1" . These tables have played an important role in astronomy.

Half a century later in Europe F.Viet found pi with only 9 correct decimal places by doing 16 doublings of the number of polygon sides. But at the same time F.Viet was the first to notice that pi can be found using the limits of some series. This discovery was of great

value, as it allowed us to calculate pi with any accuracy. Only 250 years later al-Kashi his result was surpassed.

The birthday of the number “” .

The unofficial holiday "PI Day" is celebrated on March 14, which in American format (day / date) is written as 3/14, which corresponds to an approximate value of the number of PI.

There is also an alternative version of the holiday - July 22. It's called "Approximate Pi Day". The fact is that the representation of this date as a fraction (22/7) also gives the number Pi as a result. It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who drew attention to the fact that the date and time coincide with the first digits of the number π.

Interesting facts related to the number “”

Scientists at the University of Tokyo, led by Professor Yasumasa Canada, managed to set a world record in calculating the number pi up to 12411 trillion signs. For this, a group of programmers and mathematicians needed a special program, a supercomputer and 400 hours of computer time. (Guinness Book of Records).

The German king Frederick II was so fascinated by this number that he dedicated to it ... the whole palace of Castel del Monte, in the proportions of which PI can be calculated. Now the magical palace is under the protection of UNESCO.

How to remember the first digits of the number "".

The first three digits of the number  \u003d 3.14 ... are not difficult to remember at all. And to remember more signs, there are funny sayings and poems. For example, these:

You just need to try

And remember everything as it is:

Ninety-two and six.

S.Bobrov. ”Magic Bicorn”

Anyone who learns this quatrain will always be able to name 8 digits of the number :

In the following phrases, the signs of the number  can be determined by the number of letters in each word:

“What do I know about circles? (3.1416);

“So I know the number called Pi. - Well done!"

(3,1415927);

“Learn and know in the number known behind the number the number, how to notice good luck ”

(3,14159265359)

5. The notation of the number pi

The first to introduce the notation for the ratio of the circumference of a circle to its diameter with the modern symbol pi was an English mathematician W. Johnson in 1706. As a symbol, he took the first letter of the Greek word "periphery", which means in translation "circle". Introduced W. Johnson the designation became common after the publication of the works L. Euler, who used the entered character for the first time in 1736 G.

At the end of the XVIII century. A.M. Lazhandre based on works I.G. Lambert proved that pi is irrational. Then the German mathematician F. Lindeman based on research Sh. Ermita, found a rigorous proof that this number is not only irrational, but also transcendental, i.e. cannot be the root of an algebraic equation. The search for an exact expression for pi continued after the work F. Vieta. At the beginning of the XVII century. Dutch mathematician from Cologne Ludolf van Zeulen(1540-1610) (some historians call him L. van Keulen) found 32 correct signs. Since then (publication year 1615), the value of the number p with 32 decimal places has been called the number Ludolf.

6. How to remember the number "Pi" with an accuracy of up to eleven digits

The number "Pi" is the ratio of the circumference of a circle to its diameter, it is expressed as an infinite decimal fraction. In everyday life, it is enough for us to know three signs (3.14). However, some calculations require greater accuracy.

Our ancestors did not have computers, calculators and reference books, but since the time of Peter I they have been engaged in geometric calculations in astronomy, mechanical engineering, and shipbuilding. Subsequently, electrical engineering was added here - there is the concept of "circular frequency of alternating current". To memorize the number "Pi", a couplet was invented (unfortunately, we do not know the author and the place of its first publication; but back in the late 40s of the twentieth century, Moscow schoolchildren studied according to Kiselev's geometry textbook, where it was given).

The couplet is written according to the rules of the old Russian spelling, according to which, after consonant must be placed at the end of a word "soft" or "solid" sign. Here it is, this wonderful historical couplet:

Who is joking and wishing soon

"Pi" to find out the number - already knows.

For those who are going to do accurate calculations in the future, it makes sense to remember this. So what is the number "Pi" with an accuracy of up to eleven digits? Count the number of letters in each word and write these numbers in a row (separate the first digit with a comma).

Such accuracy is already quite enough for engineering calculations. In addition to the old one, there is also a modern way of remembering, which was pointed out by a reader who identified himself as George:

So that we don't make mistakes

Must read correctly:

Three, fourteen, fifteen

Ninety-two and six.

We just have to try

And remember everything as it is:

Three, fourteen, fifteen

Ninety-two and six.

Three, fourteen, fifteen

Nine, two, six, five, three, five.

To do science

Everyone should know this.

You can just try

And keep repeating:

"Three, fourteen, fifteen,

Nine, twenty-six and five."

Well, mathematicians with the help of modern computers can calculate almost any number of digits of the number "Pi".

7. Record memorization of the number pi

Mankind has been trying to remember the signs of pi for a long time. But how to store infinity in memory? Favorite question of professional mnemonists. Many unique theories and techniques for mastering a huge amount of information have been developed. Many of them are tested on pi.

The world record set in the last century in Germany is 40,000 characters. On December 1, 2003, Alexander Belyaev set the Russian record for the values ​​of pi in Chelyabinsk. In an hour and a half, with short breaks, Alexander wrote 2,500 digits of pi on the blackboard.

Before that, it was considered a record in Russia to list 2000 characters, which was done in 1999 in Yekaterinburg. According to Alexander Belyaev, head of the Center for the Development of Figurative Memory, any of us can conduct such an experiment with our memory. It is only important to know special memorization techniques and periodically train.

Conclusion.

The number pi appears in formulas used in many fields. Physics, electrical engineering, electronics, probability theory, construction and navigation are just some of them. And it seems that just as there is no end to the signs of pi, so there is no end to the possibilities of practical application of this useful, elusive number pi.

In modern mathematics, the number pi is not only the ratio of the circumference to the diameter, it is included in a large number of different formulas.

This and other interdependencies allowed mathematicians to further understand the nature of the number pi.

The exact value of the number π in the modern world is not only of its own scientific value, but is also used for very precise calculations (for example, the orbit of a satellite, the construction of giant bridges), as well as assessing the speed and power of modern computers.

At present, the number π is associated with an incomprehensible set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this indicates a growing interest in the most important mathematical constant, the study of which has been going on for more than twenty-two centuries.

The work I did was interesting. I wanted to learn about the history of the number pi, its practical application, and I think I have achieved my goal. Summing up the work, I come to the conclusion that this topic is relevant. Many interesting facts are connected with the number π, so it is of interest to study. In my work, I became more familiar with the number - one of the eternal values ​​that mankind has been using for many centuries. Learned some aspects of its rich history. Found out why the ancient world did not know the correct ratio of circumference to diameter. I looked clearly in what ways you can get a number. Based on experiments, I calculated the approximate value of the number in various ways. Conducted processing and analysis of the results of the experiment.

Any student today should know what the number means and what the number is approximately equal to. After all, everyone has their first acquaintance with a number, using it when calculating the circumference, the area of ​​a circle occurs in the 6th grade. But, unfortunately, this knowledge remains formal for many, and after a year or two, few people remember not only that the ratio of the circumference of a circle to its diameter is the same for all circles, but even with difficulty remember the numerical value of the number equal to 3 ,fourteen.

I tried to lift the veil of the rich history of the number, which mankind has been using for many centuries. I made a presentation for my work.

The history of numbers is fascinating and mysterious. I would like to continue researching other amazing numbers in mathematics. This will be the subject of my next research studies.

Bibliography.

1. Glazer G.I. History of mathematics at school IV-VI grades. - M.: Enlightenment, 1982.

2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook - M .: Education, 1989.

3. Zhukov A.V. The ubiquitous number "pi". - M.: Editorial URSS, 2004.

4. Kympan F. The history of the number "pi". - M.: Nauka, 1971.

5. Svechnikov A.A. journey into the history of mathematics - M .: Pedagogy - Press, 1995.

6. Encyclopedia for children. T.11. Mathematics - M.: Avanta +, 1998.

Internet resources:

- http:// crow.academy.ru/ materials_/pi/history.htm

http://hab/kp.ru//daily/24123/344634/

What is the number pi we know and remember from school. It is equal to 3.1415926 and so on... It is enough for an ordinary person to know that this number is obtained by dividing the circumference of a circle by its diameter. But many people know that the number Pi appears in unexpected areas not only in mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you can see a lot of surprises among the endless series of numbers. Is it possible that Pi hides the deepest secrets of the universe?

Infinite number

The number Pi itself arises in our world as the length of a circle, the diameter of which is equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi starts like 3.1415926 and goes to infinity in rows of numbers that never repeat. The first surprising fact is that this number, used in geometry, cannot be expressed as a fraction of whole numbers. In other words, you cannot write it as a ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no such equation (polynomial) with integer coefficients, the solution of which would be Pi.

The fact that the number Pi is transcendent was proved in 1882 by the German mathematician von Lindemann. It was this proof that became the answer to the question whether it is possible to draw a square with a compass and a ruler, whose area is equal to the area of ​​a given circle. This problem is known as the search for the squaring of a circle, which has troubled mankind since ancient times. It seemed that this problem had a simple solution and was about to be revealed. But it was an incomprehensible property of pi that showed that the problem of squaring a circle has no solution.

For at least four and a half millennia, mankind has been trying to get an increasingly accurate value of pi. For example, in the Bible in the 1st Book of Kings (7:23), the number pi is taken equal to 3.

Remarkable in accuracy, the value of Pi can be found in the pyramids of Giza: the ratio of the perimeter and height of the pyramids is 22/7. This fraction gives an approximate value of Pi, equal to 3.142 ... Unless, of course, the Egyptians set such a ratio by accident. The same value already in relation to the calculation of the number Pi was received in the III century BC by the great Archimedes.

In the Ahmes Papyrus, an ancient Egyptian mathematics textbook that dates back to 1650 BC, Pi is calculated as 3.160493827.

In ancient Indian texts around the 9th century BC, the most accurate value was expressed by the number 339/108, which equaled 3.1388 ...

For almost two thousand years after Archimedes, people have been trying to find ways to calculate pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Mark Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Ariabhata, the medieval mathematician Leonardo of Pisa, known as Fibonacci, the Arab scientist Al-Khwarizmi, from whose name the word "algorithm" appeared. All of them and many other people were looking for the most accurate methods for calculating Pi, but until the 15th century they never received more than 10 digits after the decimal point due to the complexity of the calculations.

Finally, in 1400, the Indian mathematician Madhava from the Sangamagram calculated Pi with an accuracy of up to 13 digits (although he still made a mistake in the last two).

Number of signs

In the 17th century, Leibniz and Newton discovered the analysis of infinitesimal quantities, which made it possible to calculate pi more progressively - through power series and integrals. Newton himself calculated 16 decimal places, but did not mention this in his books - this became known after his death. Newton claimed that he only calculated Pi out of boredom.

At about the same time, other lesser-known mathematicians also pulled themselves up, proposing new formulas for calculating the number Pi through trigonometric functions.

For example, here is the formula used to calculate Pi by astronomy teacher John Machin in 1706: PI / 4 = 4arctg(1/5) - arctg(1/239). Using methods of analysis, Machin derived from this formula the number Pi with a hundred decimal places.

By the way, in the same 1706, the number Pi received an official designation in the form of a Greek letter: it was used by William Jones in his work on mathematics, taking the first letter of the Greek word “periphery”, which means “circle”. Born in 1707, the great Leonhard Euler popularized this designation, which is now known to any schoolchild.

Before the era of computers, mathematicians were concerned with calculating as many signs as possible. In this regard, sometimes there were curiosities. Amateur mathematician W. Shanks calculated 707 digits of pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoveries in Paris in 1937. However, nine years later, observant mathematicians found that only the first 527 characters were correctly calculated. The museum had to incur decent expenses to correct the mistake - now all the numbers are correct.

When computers appeared, the number of digits of Pi began to be calculated in completely unimaginable orders.

One of the first electronic computers ENIAC, created in 1946, which was huge and generated so much heat that the room warmed up to 50 degrees Celsius, calculated the first 2037 digits of Pi. This calculation took the car 70 hours.

As computers improved, our knowledge of pi went further and further into infinity. In 1958, 10 thousand digits of the number were calculated. In 1987, the Japanese calculated 10,013,395 characters. In 2011, Japanese researcher Shigeru Hondo passed the 10 trillion mark.

Where else can you find Pi?

So, often our knowledge of the number Pi remains at the school level, and we know for sure that this number is indispensable in the first place in geometry.

In addition to the formulas for the length and area of ​​a circle, the number Pi is used in the formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: somewhere the formulas are simple and easy to remember, and somewhere they contain very complex integrals.

Then we can meet the number Pi in mathematical formulas, where, at first glance, geometry is not visible. For example, the indefinite integral of 1/(1-x^2) is Pi.

Pi is often used in series analysis. For example, here is a simple series that converges to pi:

1/1 - 1/3 + 1/5 - 1/7 + 1/9 - .... = PI/4

Among series, pi appears most unexpectedly in the well-known Riemann zeta function. It will not be possible to tell about it in a nutshell, we will only say that someday the number Pi will help to find a formula for calculating prime numbers.

And it is absolutely amazing: Pi appears in two of the most beautiful "royal" formulas of mathematics - the Stirling formula (which helps to find the approximate value of the factorial and the gamma function) and the Euler formula (which relates as many as five mathematical constants).

However, the most unexpected discovery awaited mathematicians in probability theory. Pi is also there.

For example, the probability that two numbers are relatively prime is 6/PI^2.

Pi appears in Buffon's 18th-century needle-throwing problem: what is the probability that a needle thrown onto a sheet of paper with a pattern will cross one of the lines. If the length of the needle is L, and the distance between the lines is L, and r > L, then we can approximately calculate the value of Pi using the probability formula 2L/rPI. Just imagine - we can get Pi from random events. And by the way Pi is present in the normal probability distribution, appears in the equation of the famous Gaussian curve. Does this mean that pi is even more fundamental than just the ratio of a circle's circumference to its diameter?

We can meet Pi in physics as well. Pi appears in Coulomb's law, which describes the force of interaction between two charges, in Kepler's third law, which shows the period of revolution of a planet around the Sun, and even occurs in the arrangement of electron orbitals of a hydrogen atom. And, again, the most incredible thing is that the Pi number is hidden in the formula of the Heisenberg uncertainty principle, the fundamental law of quantum physics.

Secrets of Pi

In Carl Sagan's novel "Contact", which is based on the film of the same name, aliens inform the heroine that among the signs of Pi there is a secret message from God. From a certain position, the numbers in the number cease to be random and represent a code in which all the secrets of the Universe are recorded.

This novel actually reflected the riddle that occupies the minds of mathematicians all over the planet: is the number Pi a normal number in which the digits are scattered with the same frequency, or is there something wrong with this number. And although scientists tend to the first option (but cannot prove it), Pi looks very mysterious. A Japanese man once calculated how many times the numbers from 0 to 9 occur in the first trillion digits of pi. And I saw that the numbers 2, 4 and 8 are more common than the rest. This may be one of the hints that Pi is not quite normal, and the numbers in it are really not random.

Let's remember everything that we have read above and ask ourselves, what other irrational and transcendental number is so common in the real world?

And there are other oddities in store. For example, the sum of the first twenty digits of Pi is 20, and the sum of the first 144 digits is equal to the "number of the beast" 666.

The protagonist of the American TV series The Suspect, Professor Finch, told students that, due to the infinity of pi, any combination of numbers can occur in it, from the numbers of your date of birth to more complex numbers. For example, in the 762nd position there is a sequence of six nines. This position is called the Feynman point, after the famous physicist who noticed this interesting combination.

We also know that the number Pi contains the sequence 0123456789, but it is located on the 17,387,594,880th digit.

All this means that in the infinity of Pi you can find not only interesting combinations of numbers, but also the encoded text of "War and Peace", the Bible and even the Main Secret of the Universe, if it exists.

By the way, about the Bible. The well-known popularizer of mathematics Martin Gardner in 1966 stated that the millionth sign of the number Pi (still unknown at that time) would be the number 5. He explained his calculations by the fact that in the English version of the Bible, in the 3rd book, 14th chapter, 16 -m verse (3-14-16) the seventh word contains five letters. The million figure was received eight years later. It was number five.

Is it worth it after this to assert that the number pi is random?

The history of the number Pi begins in ancient Egypt and goes in parallel with the development of all mathematics. We meet this value for the first time within the walls of the school.

The number Pi is perhaps the most mysterious of an infinite number of others. Poems are dedicated to him, artists portray him, and a film has even been made about him. In our article, we will look at the history of development and computing, as well as the areas of application of the Pi constant in our lives.

Pi is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter. Initially, it was called the Ludolf number, and it was proposed to denote it by the letter Pi by the British mathematician Jones in 1706. After the work of Leonhard Euler in 1737, this designation became generally accepted.

The number Pi is irrational, that is, its value cannot be expressed exactly as a fraction m/n, where m and n are integers. This was first proved by Johann Lambert in 1761.

The history of the development of the number Pi has already been around 4000 years. Even the ancient Egyptian and Babylonian mathematicians knew that the ratio of the circumference to the diameter is the same for any circle and its value is a little more than three.

Archimedes proposed a mathematical method for calculating Pi, in which he inscribed in a circle and described regular polygons around it. According to his calculations, Pi was approximately equal to 22/7 ≈ 3.142857142857143.

In the 2nd century, Zhang Heng proposed two values ​​for pi: ≈ 3.1724 and ≈ 3.1622.

Indian mathematicians Aryabhata and Bhaskara found an approximate value of 3.1416.

The most accurate approximation of pi for 900 years was a calculation by the Chinese mathematician Zu Chongzhi in the 480s. He deduced that Pi ≈ 355/113 and showed that 3.1415926< Пи < 3,1415927.

Until the 2nd millennium, no more than 10 digits of Pi were calculated. Only with the development of mathematical analysis, and especially with the discovery of series, were subsequent major advances in the calculation of the constant made.

In the 1400s, Madhava was able to calculate Pi=3.14159265359. His record was broken by the Persian mathematician Al-Kashi in 1424. He in his work "Treatise on the Circumference" cited 17 digits of Pi, 16 of which turned out to be correct.

The Dutch mathematician Ludolf van Zeulen reached 20 numbers in his calculations, giving 10 years of his life for this. After his death, 15 more digits of pi were discovered in his notes. He bequeathed that these figures were carved on his tombstone.

With the advent of computers, the number Pi today has several trillion digits and this is not the limit. But, as noted in Fractals for the Classroom, for all the importance of pi, “it is difficult to find areas in scientific calculations that require more than twenty decimal places.”

In our life, the number Pi is used in many scientific fields. Physics, electronics, probability theory, chemistry, construction, navigation, pharmacology - these are just some of them that simply cannot be imagined without this mysterious number.

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According to the site Calculator888.ru - Pi number - meaning, history, who invented it.