What does pi mean? Start in science. What is Pi equal to? Methods for calculating it

If you compare circles of different sizes, you will notice the following: the sizes of different circles are proportional. This means that when the diameter of a circle increases by a certain number of times, the length of this circle also increases by the same number of times. Mathematically this can be written like this:

where C1 and C2 are the lengths of two different circles, and d1 and d2 are their diameters.
This relationship works in the presence of a coefficient of proportionality - the constant π already familiar to us. From relation (1) we can conclude: the length of a circle C is equal to the product of the diameter of this circle and a proportionality coefficient π independent of the circle:

C = π d.

This formula can also be written in another form, expressing the diameter d through the radius R of a given circle:

С = 2π R.

This formula is precisely the guide to the world of circles for seventh graders.

Since ancient times, people have tried to establish the value of this constant. For example, the inhabitants of Mesopotamia calculated the area of ​​a circle using the formula:

Where does π = 3 come from?

In ancient Egypt, the value for π was more precise. In 2000-1700 BC, a scribe called Ahmes compiled a papyrus in which we find recipes for solving various practical problems. So, for example, to find the area of ​​a circle, he uses the formula:

From what reasons did he arrive at this formula? – Unknown. Probably based on his observations, however, as other ancient philosophers did.

In the footsteps of Archimedes

Which of the two numbers is greater than 22/7 or 3.14?
- They are equal.
- Why?
- Each of them is equal to π.
A. A. Vlasov. From the Examination Card.

Some people believe that the fraction 22/7 and the number π are identically equal. But this is a misconception. In addition to the above incorrect answer in the exam (see epigraph), you can also add one very entertaining puzzle to this group. The task reads: “arrange one match so that the equality becomes true.”

The solution would be this: you need to form a “roof” for the two vertical matches on the left, using one of the vertical matches in the denominator on the right. You will get a visual image of the letter π.

Many people know that the approximation π = 22/7 was determined by the ancient Greek mathematician Archimedes. In honor of this, this approximation is often called the “Archimedean” number. Archimedes managed not only to establish an approximate value for π, but also to find the accuracy of this approximation, namely, to find a narrow numerical interval to which the value π belongs. In one of his works, Archimedes proves a chain of inequalities, which in a modern way would look like this:

can be written more simply: 3,140 909< π < 3,1 428 265...

As we can see from the inequalities, Archimedes found a fairly accurate value with an accuracy of up to 0.002. The most surprising thing is that he found the first two decimal places: 3.14... This is the value we most often use in simple calculations.

Practical use

Two people are traveling on a train:
- Look, the rails are straight, the wheels are round.
Where is the knock coming from?
- Where from? The wheels are round, but the area
circle pi er square, that’s the square that knocks!

As a rule, they become acquainted with this amazing number in the 6th-7th grade, but study it more thoroughly by the end of the 8th grade. In this part of the article we will present the basic and most important formulas that will be useful to you in solving geometric problems, but to begin with we will agree to take π as 3.14 for ease of calculation.

Perhaps the most famous formula among schoolchildren that uses π is the formula for the length and area of ​​a circle. The first, the formula for the area of ​​a circle, is written as follows:

where S is the area of ​​the circle, R is its radius, D is the diameter of the circle.

The circumference of a circle, or, as it is sometimes called, the perimeter of a circle, is calculated by the formula:

C = 2 π R = π d,

where C is the circumference, R is the radius, d is the diameter of the circle.

It is clear that the diameter d is equal to two radii R.

From the formula for circumference, you can easily find the radius of the circle:

where D is the diameter, C is the circumference, R is the radius of the circle.

These are basic formulas that every student should know. Also, sometimes it is necessary to calculate the area not of the entire circle, but only of its part - the sector. Therefore, we present it to you - a formula for calculating the area of ​​a sector of a circle. She looks like this:

where S is the area of ​​the sector, R is the radius of the circle, α is the central angle in degrees.

So mysterious 3.14

Indeed, it is mysterious. Because in honor of these magical numbers they organize holidays, make films, hold public events, write poems and much more.

For example, in 1998, a film by American director Darren Aronofsky called “Pi” was released. The film received many awards.

Every year on March 14 at 1:59:26 a.m., people interested in mathematics celebrate "Pi Day." For the holiday, people prepare a round cake, sit at a round table and discuss the number Pi, solve problems and puzzles related to Pi.

Poets also paid attention to this amazing number; an unknown person wrote:
You just have to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.

Let's have some fun!

We offer you interesting puzzles with the number Pi. Unravel the words that are encrypted below.

1. π R

2. π L

3. π k

Answers: 1. Feast; 2. File; 3. Squeak.

Pi is one of the most popular mathematical concepts. Pictures are written about him, films are made, he is played on musical instruments, poems and holidays are dedicated to him, he is sought and found in sacred texts.

Who discovered pi?

Who and when first discovered the number π still remains a mystery. It is known that the builders of ancient Babylon already made full use of it in their design. Cuneiform tablets that are thousands of years old even preserve problems that were proposed to be solved using π. True, then it was believed that π was equal to three. This is evidenced by a tablet found in the city of Susa, two hundred kilometers from Babylon, where the number π was indicated as 3 1/8.

In the process of calculating π, the Babylonians discovered that the radius of a circle as a chord enters it six times, and divided the circle into 360 degrees. And at the same time they did the same with the orbit of the sun. Thus, they decided to consider that there are 360 ​​days in a year.

In Ancient Egypt, π was equal to 3.16.
In ancient India - 3,088.
In Italy at the turn of the era, it was believed that π was equal to 3.125.

In Antiquity, the earliest mention of π refers to the famous problem of squaring the circle, that is, the impossibility of using a compass and ruler to construct a square whose area is equal to the area of ​​a certain circle. Archimedes equated π to the fraction 22/7.

The closest people to the exact value of π came in China. It was calculated in the 5th century AD. e. famous Chinese astronomer Tzu Chun Zhi. π was calculated quite simply. It was necessary to write the odd numbers twice: 11 33 55, and then, dividing them in half, place the first in the denominator of the fraction, and the second in the numerator: 355/113. The result agrees with modern calculations of π up to the seventh digit.

Why π – π?

Now even schoolchildren know that the number π is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter and is equal to π 3.1415926535 ... and then after the decimal point - to infinity.

The number acquired its designation π in a complex way: first, in 1647, the mathematician Outrade used this Greek letter to describe the length of a circle. He took the first letter of the Greek word περιφέρεια - “periphery”. In 1706, the English teacher William Jones in his work “Review of the Achievements of Mathematics” already called the ratio of the circumference of a circle to its diameter by the letter π. And the name was cemented by the 18th century mathematician Leonard Euler, before whose authority the rest bowed their heads. So π became π.

Uniqueness of the number

Pi is a truly unique number.

1. Scientists believe that the number of digits in the number π is infinite. Their sequence is not repeated. Moreover, no one will ever be able to find repetitions. Since the number is infinite, it can contain absolutely everything, even a Rachmaninoff symphony, the Old Testament, your phone number and the year in which the Apocalypse will occur.

2. π is associated with chaos theory. Scientists came to this conclusion after creating Bailey's computer program, which showed that the sequence of numbers in π is absolutely random, which is consistent with the theory.

3. It is almost impossible to calculate the number completely - it would take too much time.

4. π is an irrational number, that is, its value cannot be expressed as a fraction.

5. π – transcendental number. It cannot be obtained by performing any algebraic operations on integers.

6. Thirty-nine decimal places in the number π are enough to calculate the length of the circle encircling known cosmic objects in the Universe, with an error of the radius of a hydrogen atom.

7. The number π is associated with the concept of the “golden ratio”. In the process of measuring the Great Pyramid of Giza, archaeologists discovered that its height is related to the length of its base, just as the radius of a circle is related to its length.

Records related to π

In 2010, Yahoo mathematician Nicholas Zhe was able to calculate two quadrillion decimal places (2x10) in the number π. It took 23 days, and the mathematician needed many assistants who worked on thousands of computers, united using distributed computing technology. The method made it possible to perform calculations at such a phenomenal speed. To calculate the same thing on a single computer would take more than 500 years.

In order to simply write all this down on paper, you would need a paper tape more than two billion kilometers long. If you expand such a record, its end will go beyond the solar system.

Chinese Liu Chao set a record for memorizing the sequence of digits of the number π. Within 24 hours and 4 minutes, Liu Chao said 67,890 decimal places without making a single mistake.

π has many fans. It is played on musical instruments, and it turns out that it “sounds” excellent. They remember it and come up with various techniques for this. For fun, they download it to their computer and brag to each other about who has downloaded the most. Monuments are erected to him. For example, there is such a monument in Seattle. It is located on the steps in front of the Museum of Art.

π is used in decorations and interior design. Poems are dedicated to him, he is looked for in holy books and at excavations. There is even a “Club π”.
In the best traditions of π, not one, but two whole days a year are dedicated to the number! The first time π Day is celebrated is March 14th. You need to congratulate each other at exactly 1 hour, 59 minutes, 26 seconds. Thus, the date and time correspond to the first digits of the number - 3.1415926.

For the second time, the π holiday is celebrated on July 22. This day is associated with the so-called “approximate π”, which Archimedes wrote down as a fraction.
Usually on this day, students, schoolchildren and scientists organize funny flash mobs and actions. Mathematicians, having fun, use π to calculate the laws of a falling sandwich and give each other comic rewards.
And by the way, π can actually be found in the holy books. For example, in the Bible. And there the number π is equal to... three.

Introduction

The article contains mathematical formulas, so to read, go to the site to display them correctly. The number \(\pi\) has a rich history. This constant denotes the ratio of the circumference of a circle to its diameter.

In science, the number \(\pi \) is used in any calculations involving circles. Starting from the volume of a can of soda, to the orbits of satellites. And not just circles. Indeed, in the study of curved lines, the number \(\pi \) helps to understand periodic and oscillatory systems. For example, electromagnetic waves and even music.

In 1706, in the book A New Introduction to Mathematics by the British scientist William Jones (1675-1749), the letter of the Greek alphabet \(\pi\) was first used to represent the number 3.141592.... This designation comes from the initial letter of the Greek words περιϕερεια - circle, periphery and περιµετρoς - perimeter. The designation became generally accepted after the work of Leonhard Euler in 1737.

Geometric period

The constancy of the ratio of the length of any circle to its diameter has been noticed for a long time. The inhabitants of Mesopotamia used a rather rough approximation of the number \(\pi\). As follows from ancient problems, they use the value \(\pi ≈ 3\) in their calculations.

A more precise value for \(\pi\) was used by the ancient Egyptians. In London and New York, two pieces of ancient Egyptian papyrus are kept, which are called the “Rinda papyrus”. The papyrus was compiled by the scribe Armes sometime between 2000-1700. BC. Armes wrote in his papyrus that the area of ​​a circle with radius \(r\) is equal to the area of ​​a square with a side equal to \(\frac(8)(9) \) of the diameter of the circle \(\frac(8 )(9) \cdot 2r \), that is, \(\frac(256)(81) \cdot r^2 = \pi r^2 \). Hence \(\pi = 3.16\).

The ancient Greek mathematician Archimedes (287-212 BC) was the first to put the problem of measuring a circle on a scientific basis. He received a score of \(3\frac(10)(71)< \pi < 3\frac{1}{7}\), рассмотрев отношение периметров вписанного и описанного 96-угольника к диаметру окружности. Архимед выразил приближение числа \(\pi \) в виде дроби \(\frac{22}{7}\), которое до сих называется архимедовым числом.

The method is quite simple, but in the absence of ready-made tables of trigonometric functions, extraction of roots will be required. In addition, the approximation converges to \(\pi \) very slowly: with each iteration the error decreases only fourfold.

Analytical period

Despite this, until the mid-17th century, all attempts by European scientists to calculate the number \(\pi\) boiled down to increasing the sides of the polygon. For example, the Dutch mathematician Ludolf van Zeijlen (1540-1610) calculated the approximate value of the number \(\pi\) accurate to 20 decimal digits.

It took him 10 years to calculate. By doubling the number of sides of inscribed and circumscribed polygons using Archimedes' method, he arrived at \(60 \cdot 2^(29) \) - a triangle in order to calculate \(\pi \) with 20 decimal places.

After his death, 15 more exact digits of the number \(\pi\) were discovered in his manuscripts. Ludolf bequeathed that the signs he found be carved on his tombstone. In his honor, the number \(\pi\) was sometimes called the "Ludolf number" or "Ludolf constant".

One of the first to introduce a method different from that of Archimedes was François Viète (1540-1603). He came to the result that a circle whose diameter is equal to one has an area:

\[\frac(1)(2 \sqrt(\frac(1)(2)) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1 )(2)) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt (\frac(1)(2) \cdots )))) \]

On the other hand, the area is \(\frac(\pi)(4)\). By substituting and simplifying the expression, we can obtain the following infinite product formula for calculating the approximate value of \(\frac(\pi)(2)\):

\[\frac(\pi)(2) = \frac(2)(\sqrt(2)) \cdot \frac(2)(\sqrt(2 + \sqrt(2))) \cdot \frac(2 )(\sqrt(2+ \sqrt(2 + \sqrt(2)))) \cdots \]

The resulting formula is the first exact analytical expression for the number \(\pi\). In addition to this formula, Viet, using the method of Archimedes, gave, using inscribed and circumscribed polygons, starting with a 6-gon and ending with a polygon with \(2^(16) \cdot 6 \) sides, an approximation of the number \(\pi \) with 9 with the right signs.

The English mathematician William Brounker (1620-1684), using continued fraction, obtained the following results for calculating \(\frac(\pi)(4)\):

\[\frac(4)(\pi) = 1 + \frac(1^2)(2 + \frac(3^2)(2 + \frac(5^2)(2 + \frac(7^2) )(2 + \frac(9^2)(2 + \frac(11^2)(2 + \cdots )))))) \]

This method of calculating the approximation of the number \(\frac(4)(\pi)\) requires quite a lot of calculations to get even a small approximation.

The values ​​obtained as a result of substitution are either greater or less than the number \(\pi\), and each time they are closer to the true value, but to obtain the value 3.141592 it will be necessary to perform quite large calculations.

Another English mathematician John Machin (1686-1751) in 1706, to calculate the number \(\pi\) with 100 decimal places, used the formula derived by Leibniz in 1673 and applied it as follows:

\[\frac(\pi)(4) = 4 arctg\frac(1)(5) - arctg\frac(1)(239) \]

The series converges quickly and with its help you can calculate the number \(\pi \) with great accuracy. These types of formulas have been used to set several records during the computer era.

In the 17th century with the beginning of the period of variable-value mathematics, a new stage in the calculation of \(\pi\) began. The German mathematician Gottfried Wilhelm Leibniz (1646-1716) in 1673 found a decomposition of the number \(\pi\), in general it can be written as the following infinite series:

\[ \pi = 1 — 4(\frac(1)(3) + \frac(1)(5) — \frac(1)(7) + \frac(1)(9) — \frac(1) (11) + \cdots) \]

The series is obtained by substituting x = 1 into \(arctg x = x - \frac(x^3)(3) + \frac(x^5)(5) - \frac(x^7)(7) + \frac (x^9)(9) — \cdots\)

Leonhard Euler develops Leibniz's idea in his works on the use of series for arctan x in calculating the number \(\pi\). The treatise "De variis modis circuli quadraturam numeris proxime exprimendi" (On various methods of expressing the squaring of the circle by approximate numbers), written in 1738, discusses methods for improving the calculations using Leibniz's formula.

Euler writes that the series for the arctangent will converge faster if the argument tends to zero. For \(x = 1\), the convergence of the series is very slow: to calculate with an accuracy of 100 digits it is necessary to add \(10^(50)\) terms of the series. You can speed up calculations by decreasing the value of the argument. If we take \(x = \frac(\sqrt(3))(3)\), then we get the series

\[ \frac(\pi)(6) = artctg\frac(\sqrt(3))(3) = \frac(\sqrt(3))(3)(1 — \frac(1)(3 \cdot 3) + \frac(1)(5 \cdot 3^2) — \frac(1)(7 \cdot 3^3) + \cdots) \]

According to Euler, if we take 210 terms of this series, we will get 100 correct digits of the number. The resulting series is inconvenient because it is necessary to know a fairly accurate value of the irrational number \(\sqrt(3)\). Euler also used in his calculations expansions of arctangents into the sum of arctangents of smaller arguments:

\[where x = n + \frac(n^2-1)(m-n), y = m + p, z = m + \frac(m^2+1)(p) \]

Not all the formulas for calculating \(\pi\) that Euler used in his notebooks were published. In published papers and notebooks, he considered 3 different series for calculating the arctangent, and also made many statements regarding the number of summable terms required to obtain an approximate value of \(\pi\) with a given accuracy.

In subsequent years, refinements to the value of the number \(\pi\) occurred faster and faster. For example, in 1794, Georg Vega (1754-1802) already identified 140 signs, of which only 136 turned out to be correct.

Computing period

The 20th century was marked by a completely new stage in the calculation of the number \(\pi\). Indian mathematician Srinivasa Ramanujan (1887-1920) discovered many new formulas for \(\pi\). In 1910, he obtained a formula for calculating \(\pi\) through the arctangent expansion in a Taylor series:

\[\pi = \frac(9801)(2\sqrt(2) \sum\limits_(k=1)^(\infty) \frac((1103+26390k) \cdot (4k){(4\cdot99)^{4k} (k!)^2}} .\]!}

At k=100, an accuracy of 600 correct digits of the number \(\pi\) is achieved.

The advent of computers made it possible to significantly increase the accuracy of the obtained values ​​in a shorter time. In 1949, in just 70 hours, using ENIAC, a group of scientists led by John von Neumann (1903-1957) obtained 2037 decimal places for the number \(\pi\). In 1987, David and Gregory Chudnovsky obtained a formula with which they were able to set several records in calculating \(\pi\):

\[\frac(1)(\pi) = \frac(1)(426880\sqrt(10005)) \sum\limits_(k=1)^(\infty) \frac((6k)!(13591409+545140134k ))((3k)!(k!)^3(-640320)^(3k)).\]

Each member of the series gives 14 digits. In 1989, 1,011,196,691 decimal places were obtained. This formula is well suited for calculating \(\pi \) on personal computers. Currently, the brothers are professors at the Polytechnic Institute of New York University.

An important recent development was the discovery of the formula in 1997 by Simon Plouffe. It allows you to extract any hexadecimal digit of the number \(\pi\) without calculating the previous ones. The formula is called the “Bailey-Borwain-Plouffe Formula” in honor of the authors of the article where the formula was first published. It looks like this:

\[\pi = \sum\limits_(k=1)^(\infty) \frac(1)(16^k) (\frac(4)(8k+1) — \frac(2)(8k+4 ) - \frac(1)(8k+5) - \frac(1)(8k+6)) .\]

In 2006, Simon, using PSLQ, came up with some nice formulas for calculating \(\pi\). For example,

\[ \frac(\pi)(24) = \sum\limits_(n=1)^(\infty) \frac(1)(n) (\frac(3)(q^n - 1) - \frac (4)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]

\[ \frac(\pi^3)(180) = \sum\limits_(n=1)^(\infty) \frac(1)(n^3) (\frac(4)(q^(2n) — 1) — \frac(5)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]

where \(q = e^(\pi)\). In 2009, Japanese scientists, using the T2K Tsukuba System supercomputer, obtained the number \(\pi\) with 2,576,980,377,524 decimal places. The calculations took 73 hours 36 minutes. The computer was equipped with 640 quad-core AMD Opteron processors, which provided performance of 95 trillion operations per second.

The next achievement in calculating \(\pi\) belongs to the French programmer Fabrice Bellard, who at the end of 2009, on his personal computer running Fedora 10, set a record by calculating 2,699,999,990,000 decimal places of the number \(\pi\). Over the past 14 years, this is the first world record that was set without the use of a supercomputer. For high performance, Fabrice used the Chudnovsky brothers' formula. In total, the calculation took 131 days (103 days of calculations and 13 days of verification of the result). Bellar's achievement showed that such calculations do not require a supercomputer.

Just six months later, Francois's record was broken by engineers Alexander Yi and Singer Kondo. To set a record of 5 trillion decimal places of \(\pi\), a personal computer was also used, but with more impressive characteristics: two Intel Xeon X5680 processors at 3.33 GHz, 96 GB of RAM, 38 TB of disk memory and operating system Windows Server 2008 R2 Enterprise x64. For calculations, Alexander and Singer used the formula of the Chudnovsky brothers. The calculation process took 90 days and 22 TB of disk space. In 2011, they set another record by calculating 10 trillion decimal places for the number \(\pi\). The calculations took place on the same computer on which their previous record was set and took a total of 371 days. At the end of 2013, Alexander and Singerou improved the record to 12.1 trillion digits of the number \(\pi\), which took them only 94 days to calculate. This performance improvement is achieved by optimizing software performance, increasing the number of processor cores, and significantly improving software fault tolerance.

The current record is that of Alexander Yee and Singer Kondo, which is 12.1 trillion decimal places \(\pi\).

Thus, we looked at methods for calculating the number \(\pi\) used in ancient times, analytical methods, and also looked at modern methods and records for calculating the number \(\pi\) on computers.

List of sources

1. Zhukov A.V. The ubiquitous number Pi - M.: Publishing house LKI, 2007 - 216 p.
2. F.Rudio. On the squaring of the circle, with the application of a history of the issue compiled by F. Rudio. / Rudio F. – M.: ONTI NKTP USSR, 1936. – 235c.
3. Arndt, J. Pi Unleashed / J. Arndt, C. Haenel. – Springer, 2001. – 270p.
4. Shukhman, E.V. Approximate calculation of Pi using the series for arctan x in published and unpublished works of Leonhard Euler / E.V. Shukhman. — History of science and technology, 2008 – No. 4. – P. 2-17.
5. Euler, L. De variis modis circuli quadraturam numeris proxime exprimendi/ Commentarii academiae scientiarum Petropolitanae. 1744 – Vol.9 – 222-236p.
6. Shumikhin, S. Number Pi. A history of 4000 years / S. Shumikhin, A. Shumikhina. - M.: Eksmo, 2011. - 192 p.
7. Borwein, J.M. Ramanujan and the number Pi. / Borwein, J.M., Borwein P.B. In the world of science. 1988 – No. 4. – pp. 58-66.
8. Alex Yee. Number world. Access mode: numberworld.org

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On March 14, a very unusual holiday is celebrated all over the world - Pi Day. Everyone has known it since school. Students are immediately explained that the number Pi is a mathematical constant, the ratio of the circumference of a circle to its diameter, which has an infinite value. It turns out that there are many interesting facts associated with this number.

1. The history of numbers goes back more than one thousand years, almost as long as the science of mathematics has existed. Of course, the exact value of the number was not immediately calculated. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, a more accurate measurement was required. By the way, the number existed, but it received a letter designation only at the beginning of the 18th century (1706) and comes from the initial letters of two Greek words meaning “circle” and “perimeter”. The letter “π” was given to the number by the mathematician Jones, and it became firmly established in mathematics already in 1737.

2. In different eras and among different peoples, the number Pi had different meanings. For example, in Ancient Egypt it was equal to 3.1604, among the Hindus it acquired a value of 3.162, and the Chinese used a number equal to 3.1459. Over time, π was calculated more and more accurately, and when computing technology, that is, a computer, appeared, it began to number more than 4 billion characters.

3. There is a legend, or rather experts believe, that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were wrong. A similar version exists regarding the Temple of Solomon.

4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through law. In 1897, the state of Indiana prepared a bill. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented the mistake. In particular, Professor Perdue, who was present at the legislative meeting, spoke out against the bill.

5. It is interesting that several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after the American physicist. Richard Feynman once gave a lecture and stunned the audience with a remark. He said he wanted to memorize the digits of Pi up to six nines, only to say "nine" six times at the end of the story, implying that its meaning was rational. When in fact it is irrational.

6. Mathematicians around the world do not stop conducting research related to the number Pi. It is literally shrouded in some mystery. Some theorists even believe that it contains universal truth. To exchange knowledge and new information about Pi, a Pi Club was organized. It’s not easy to join; you need to have an extraordinary memory. Thus, those wishing to become a member of the club are examined: a person must recite from memory as many signs of the number Pi as possible.

7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with entire texts. In them, words have the same number of letters as the corresponding number after the decimal point. To make it even easier to remember such a long number, they compose poems according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and intelligence. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) of the first digits of Pi.

8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk managed to recite by heart only two and a half thousand numbers after the decimal point of Pi.

"Pi" in perspective

9. Pi Day has been celebrated for more than a quarter of a century, since 1988. One day, a physicist from the popular science museum in San Francisco, Larry Shaw, noticed that March 14, when written, coincides with the number Pi. In the date, the month and day form 3.14.

10. Pi Day is celebrated not exactly in an original way, but in a fun way. Of course, scientists involved in exact sciences do not miss it. For them, this is a way not to break away from what they love, but at the same time relax. On this day, people gather and prepare various delicacies with the image of Pi. There is especially room for pastry chefs to roam. They can make cakes with pi written on them and cookies with similar shapes. After tasting the delicacies, mathematicians arrange various quizzes.

11. There is an interesting coincidence. On March 14, the great scientist Albert Einstein, who, as we know, created the theory of relativity, was born. Be that as it may, physicists can also join in the celebration of Pi Day.

For calculating any large number of signs of pi, the previous method is no longer suitable. But there are a large number of sequences that converge to Pi much faster. Let us use, for example, the Gauss formula:

The proof of this formula is not difficult, so we will omit it.

Source code of the program, including "long arithmetic"

The program calculates NbDigits of the first digits of Pi. The function for calculating arctan is called arccot, since arctan(1/p) = arccot(p), but the calculation is carried out according to the Taylor formula specifically for the arctangent, namely arctan(x) = x - x 3 /3 + x 5 /5 - . .. x=1/p, which means arccot(x) = 1/p - 1 / p 3 / 3 + ... Calculations occur recursively: the previous element of the sum is divided and gives the next one.