What is another name for the number pi? What is the number PI and what does it mean? Brief history of π calculations

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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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.

What is Pi equal to? 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 of mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you will notice many surprising things among the endless series of numbers. Is it possible that Pi is hiding the deepest secrets of the universe?

Infinite number

The number Pi itself appears in our world as the length of a circle whose diameter is equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi begins as 3.1415926 and goes to infinity in rows of numbers that are never repeated. 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 the ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no equation (polynomial) with integer coefficients whose solution would be the number Pi.

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

For at least four and a half millennia, humanity has been trying to obtain an increasingly accurate value for Pi. For example, in the Bible in the Third Book of Kings (7:23), the number Pi is taken to be 3.

The Pi value of remarkable accuracy can be found in the Giza pyramids: 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 this ratio by accident. The same value was already obtained in relation to the calculation of the number Pi in the 3rd century BC by the great Archimedes.

In the Papyrus of Ahmes, 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 was equal to 3.1388...

For almost two thousand years after Archimedes, people tried to find ways to calculate Pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Marcus Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Aryabhata, 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 got more than 10 decimal places due to the complexity of the calculations.

Finally, in 1400, the Indian mathematician Madhava from Sangamagram calculated Pi with an accuracy of 13 digits (although he was still mistaken 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 it in his books - this became known after his death. Newton claimed that he calculated Pi purely out of boredom.

Around the same time, other lesser-known mathematicians also came forward and proposed new formulas for calculating the number Pi through trigonometric functions.

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

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

Before the era of computers, mathematicians focused on calculating as many signs as possible. In this regard, sometimes funny things arose. Amateur mathematician W. Shanks calculated 707 digits of Pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoverys in Paris in 1937. However, nine years later, observant mathematicians discovered that only the first 527 characters were correctly calculated. The museum had to incur significant expenses to correct the error - now all the figures 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, was enormous in size 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 machine 70 hours.

As computers improved, our knowledge of Pi moved 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 exceeded the 10 trillion character mark.

Where else can you meet Pi?

So, often our knowledge about the number Pi remains at the school level, and we know for sure that this number is irreplaceable primarily in geometry.

In addition to formulas for the length and area of ​​a circle, the number Pi is used in formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: in some places the formulas are simple and easy to remember, but in others 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 equal to 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 the series, Pi appears most unexpectedly in the famous Riemann zeta function. It’s impossible to talk about it in a nutshell, let’s just say that someday the number Pi will help find a formula for calculating prime numbers.

And absolutely surprisingly: Pi appears in two of the most beautiful “royal” formulas of mathematics - Stirling’s formula (which helps to find the approximate value of the factorial and gamma function) and Euler’s formula (which connects as many as five mathematical constants).

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

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

Pi appears in Buffon's needle-throwing problem, formulated in the 18th century: what is the probability that a needle thrown onto a lined piece of paper 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 simply the ratio of circumference to diameter?

We can also meet Pi in physics. 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 appears in the arrangement of the electron orbitals of the hydrogen atom. And what is again most incredible is that the number Pi 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, on which the film of the same name is based, aliens tell 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 written.

This novel actually reflected a mystery that has occupied the minds of mathematicians all over the world: is Pi a normal number in which the digits are scattered with equal frequency, or is there something wrong with this number? And although scientists are inclined to the first option (but cannot prove it), the number Pi looks very mysterious. A Japanese man once calculated how many times the numbers 0 to 9 occur in the first trillion digits of Pi. And I saw that the numbers 2, 4 and 8 were more common than the others. This may be one of the hints that Pi is not entirely normal, and the numbers in it are indeed not random.

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

And there are more 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 main character of the American TV series “Suspect,” Professor Finch, told students that due to the infinity of the number Pi, any combination of numbers can be found in it, ranging from the numbers of your date of birth to more complex numbers. For example, at position 762 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 at the 17,387,594,880th digit.

All this means that in the infinity of the number Pi one 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 such exists.

By the way, about the Bible. The famous popularizer of mathematics, Martin Gardner, stated in 1966 that the millionth digit of Pi (at that time still unknown) 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 verse (3-14-16) the seventh word contains five letters. The millionth figure was reached eight years later. It was the number five.

Is it worth asserting after this that the number Pi is random?

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.