Pi attitude. Start in science

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 old 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.

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?

I never thought about the story of the origin of Pi. I read quite interesting facts about Leibniz and Newton. Newton calculated 16 decimal places but didn't tell in his book. Thanks for the good article.

Once I read on a forum about magic that the number PI has not only a magical meaning, but also a ritual one. Many rituals are associated with this number and have been used by magicians since ancient times of the discovery of this number.

the sum of the first twenty digits of pi is 20… Is this serious? In a binary system, right?

Reply
1. 100 is not the sum of the first 20 digits, but 20 decimal places.
  
  Reply

with diameter = 1, the circumference = pi, and, therefore, the circle will never close!

Reply

NUMBER p - the ratio of the circumference of a circle to its diameter, - the value is constant and does not depend on the size of the circle. The number expressing this relationship is usually denoted by the Greek letter 241 (from "perijereia" - circle, periphery). This designation became common after the work of Leonhard Euler, referring to 1736, but it was first used by William Jones (1675–1749) in 1706. Like any irrational number, it is represented by an infinite non-periodic decimal fraction:

p= 3.141592653589793238462643… The needs of practical calculations relating to circles and round bodies forced us to search for 241 approximations using rational numbers already in ancient times. Information that the circumference is exactly three times longer than the diameter is found in the cuneiform tablets of the Ancient Mesopotamia. Same number value p there is also in the text of the Bible: “And he made a sea of cast copper, from end to end it was ten cubits, completely round, five cubits high, and a string of thirty cubits hugged it around” (1 Kings 7.23). So did the ancient Chinese. But already in 2 thousand BC. the ancient Egyptians used a more accurate value for the number 241, which is obtained from the formula for the area of a circle of diameter d:

This rule from the 50th problem of the Rhind papyrus corresponds to the value 4(8/9) 2 » 3.1605. The Rhinda Papyrus, found in 1858, is named after its first owner, it was copied by the scribe Ahmes around 1650 BC, the author of the original is unknown, it is only established that the text was created in the second half of the 19th century. BC. Although how the Egyptians got the formula itself is not clear from the context. In the so-called Moscow papyrus, which was copied by a certain student between 1800 and 1600 BC. from an older text, circa 1900 BC, there is another interesting problem about calculating the surface of a basket "with an opening of 4½". It is not known what shape the basket was, but all researchers agree that here for the number p the same approximate value 4(8/9) 2 is taken.

In order to understand how the ancient scientists obtained this or that result, one should try to solve the problem using only the knowledge and methods of calculations of that time. This is exactly what researchers of ancient texts do, but the solutions they manage to find are not necessarily “the same ones”. Very often, several solutions are offered for one task, everyone can choose according to their taste, but no one can say that it was used in antiquity. Regarding the area of a circle, the hypothesis of A.E. Raik, the author of numerous books on the history of mathematics, seems plausible: the area of a circle of diameter d is compared with the area of the square described around it, from which small squares with sides and are removed in turn (Fig. 1). In our notation, the calculations will look like this: in the first approximation, the area of the circle S equal to the difference between the area of a square with a side d and the total area of four small squares BUT with a party d:

This hypothesis is supported by similar calculations in one of the problems of the Moscow Papyrus, where it is proposed to calculate

From the 6th c. BC. mathematics developed rapidly in ancient Greece. It was the ancient Greek geometers who strictly proved that the circumference of a circle is proportional to its diameter ( l = 2p R; R is the radius of the circle, l - its length), and the area of a circle is half the product of the circumference and radius:

S = ½ l R = p R 2 .

This evidence is attributed to Eudoxus of Cnidus and Archimedes.

In the 3rd century BC. Archimedes in writing About measuring a circle calculated the perimeters of regular polygons inscribed in a circle and described around it (Fig. 2) - from a 6- to a 96-gon. Thus he established that the number p lies between 3 10/71 and 3 1/7, i.e. 3.14084< p < 3,14285. Последнее значение до сих пор используется при расчетах, не требующих особой точности. Более точное приближение 3 17/120 (p» 3.14166) was found by the famous astronomer, the creator of trigonometry, Claudius Ptolemy (2nd century), but it did not come into use.

Indians and Arabs believed that p= . This value is also given by the Indian mathematician Brahmagupta (598 - ca. 660). In China, scientists in the 3rd century. used the value 3 7/50, which is worse than the approximation of Archimedes, but in the second half of the 5th c. Zu Chun Zhi (c. 430 - c. 501) received for p approximation 355/113 ( p» 3.1415927). It remained unknown to Europeans and was again found by the Dutch mathematician Adrian Antonis only in 1585. This approximation gives an error only in the seventh decimal place.

The search for a more accurate approximation p continued further. For example, al-Kashi (first half of the 15th century) in Treatise on the Circle(1427) computed 17 decimal places p. In Europe, the same meaning was found in 1597. To do this, he had to calculate the side of a regular 800 335 168-gon. The Dutch scientist Ludolph Van Zeilen (1540–1610) found 32 correct decimal places for it (published posthumously in 1615), this approximation is called the Ludolf number.

Number p appears not only in solving geometric problems. Since the time of F. Vieta (1540–1603), the search for the limits of some arithmetic sequences compiled according to simple laws has led to the same number p. For this reason, in determining the number p almost all famous mathematicians took part: F. Viet, H. Huygens, J. Wallis, G. V. Leibniz, L. Euler. They received various expressions for 241 in the form of an infinite product, the sum of a series, an infinite fraction.

For example, in 1593 F. Viet (1540–1603) derived the formula

In 1658 the Englishman William Brounker (1620–1684) found a representation of the number p as an infinite continued fraction

however, it is not known how he arrived at this result.

In 1665 John Wallis (1616–1703) proved that

This formula bears his name. For the practical determination of the number 241, it is of little use, but is useful in various theoretical reasoning. It entered the history of science as one of the first examples of infinite works.

Gottfried Wilhelm Leibniz (1646–1716) established the following formula in 1673:

expressing number p/4 as the sum of the series. However, this series converges very slowly. To calculate p accurate to ten digits, it would be necessary, as Isaac Newton showed, to find the sum of 5 billion numbers and spend about a thousand years of continuous work on this.

London mathematician John Machin (1680–1751) in 1706, applying the formula

got the expression

which is still considered one of the best for approximate calculation p. It only takes a few hours of manual counting to find the same ten exact decimal places. John Machin himself calculated p with 100 correct characters.

Using the same row for arctg x and formulas

number value p received on a computer with an accuracy of one hundred thousand decimal places. Such calculations are of interest in connection with the concept of random and pseudo-random numbers. Statistical processing of an ordered set of a specified number of characters p shows that it has many of the features of a random sequence.

There are some fun ways to remember a number p more precisely than just 3.14. For example, having learned the following quatrain, you can easily name seven decimal places p:

You just need to try

And remember everything as it is:

Three, fourteen, fifteen

ninety two and six.

(S.Bobrov Magic Bicorn)

Counting the number of letters in each word of the following phrases also gives the value of the number p:

"What do I know about circles?" ( p» 3.1416). This proverb was suggested by Ya.I. Perelman.

“So I know the number called Pi. - Well done!" ( p» 3.1415927).

“Learn and know in the number known behind the number the number, how to notice good luck” ( p» 3.14159265359).

The teacher of one of the Moscow schools came up with the line: “I know this and remember it perfectly,” and his student composed a funny continuation: “Many signs are superfluous to me, in vain.” This couplet allows you to define 12 digits.

And this is what 101 digits of a number look like p without rounding

3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679.

Nowadays, with the help of a computer, the value of a number p calculated with millions of correct digits, but such precision is not needed in any calculations. But the possibility of analytical determination of the number ,

In the last formula, the numerator contains all prime numbers, and the denominators differ from them by one, and the denominator is greater than the numerator if it has the form 4 n+ 1, and less otherwise.

Although since the end of the 16th century, i.e. since the very concepts of rational and irrational numbers were formed, many scientists have been convinced that p- the number is irrational, but only in 1766 the German mathematician Johann Heinrich Lambert (1728–1777), based on the relationship between the exponential and trigonometric functions discovered by Euler, strictly proved this. Number p cannot be represented as a simple fraction, no matter how large the numerator and denominator are.

In 1882, professor at the University of Munich, Carl Louis Ferdinand Lindemann (1852–1939), using the results obtained by the French mathematician C. Hermite, proved that p- a transcendental number, i.e. it is not the root of any algebraic equation a n x n + a n– 1 x n– 1 + … + a 1 x + a 0 = 0 with integer coefficients. This proof put an end to the history of the oldest mathematical problem of squaring a circle. For thousands of years, this problem has not yielded to the efforts of mathematicians, the expression "squaring the circle" has become synonymous with an unsolvable problem. And the whole thing turned out to be in the transcendental nature of the number p.

In memory of this discovery, a bust of Lindemann was erected in the hall in front of the mathematical auditorium of the University of Munich. On the pedestal under his name is a circle crossed by a square of equal area, inside which the letter is inscribed p.

Marina Fedosova

Introduction

The article contains mathematical formulas, so for reading go to the site for their correct display. 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 calculation where there are 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 used for the first time to denote the number 3.141592.... This designation comes from the initial letter of the Greek words περιϕερεια - circle, periphery and περιµετρoς - perimeter. The generally accepted designation became 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 parts of an ancient Egyptian papyrus are kept, which is called the "Rhinda Papyrus". The papyrus was compiled by the scribe Armes between about 2000-1700 BC. BC. Armes wrote in his papyrus that the area of a circle with a radius \(r\) is equal to the area of a square with a side equal to \(\frac(8)(9) \) from the diameter of the circle \(\frac(8 )(9) \cdot 2r \), i.e. \(\frac(256)(81) \cdot r^2 = \pi r^2 \). Hence \(\pi = 3,16\).

The ancient Greek mathematician Archimedes (287-212 BC) first set the task of measuring a circle on a scientific basis. He got the score \(3\frac(10)(71)< \pi < 3\frac{1}{7}\), 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, root extraction will be required. In addition, the approximation to \(\pi \) converges very slowly: with each iteration, the error only decreases by a factor of four.

Analytical period

Despite this, until the middle of the 17th century, all attempts by European scientists to calculate the number \(\pi \) were reduced to increasing the sides of the polygon. For example, the Dutch mathematician Ludolf van Zeilen (1540-1610) calculated the approximate value of the number \(\pi \) with an accuracy of 20 decimal digits.

It took him 10 years to figure it out. By doubling the number of sides of the inscribed and circumscribed polygons according to the method of Archimedes, he came up with \(60 \cdot 2^(29) \) - a square in order to calculate \(\pi \) with 20 decimal places.

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

One of the first to introduce a method different from that of Archimedes was François Viet (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 \(\frac(\pi)(4) \). Substituting and simplifying the expression, we can obtain the following infinite product formula for calculating the approximate value \(\frac(\pi)(2) \):

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

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 with the help of 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 correct signs.

The English mathematician William Brounker (1620-1684) used the continued fraction to calculate \(\frac(\pi)(4)\) as follows:

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

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

The values obtained as a result of the substitution are either greater or less than the number \(\pi \), and each time closer to the true value, but getting the value 3.141592 will require quite a large calculation.

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

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

The series converges quickly and can be used to calculate the number \(\pi \) with great accuracy. Formulas of this type were used to set several records in the computer age.

In the 17th century with the beginning of the period of mathematics of variable magnitude, a new stage began in the calculation of \(\pi \). The German mathematician Gottfried Wilhelm Leibniz (1646-1716) in 1673 found the expansion of the number \(\pi \), in general form 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 the idea of Leibniz in his work on the use of series for arctg x when calculating the number \(\pi \). The treatise "De variis modis circuli quadraturam numeris proxime exprimendi" (On the various methods of expressing the squaring of a circle by approximate numbers), written in 1738, discusses methods for improving calculations using the Leibniz formula.

Euler writes that the arc tangent series 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 up to 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 get 100 correct digits of the number. The resulting series is inconvenient, because it is necessary to know a sufficiently precise value of the irrational number \(\sqrt(3)\). Also, in his calculations, Euler used expansions of arc tangents into the sum of arc tangents of smaller arguments:

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

Far from all the formulas for calculating \(\pi \) that Euler used in his notebooks have been published. In published works and notebooks, he considered 3 different series for calculating the arc tangent, and also made many statements regarding the number of summable terms needed to obtain an approximate value \(\pi \) with a given accuracy.

In subsequent years, the refinement of the value of the number \(\pi \) happened faster and faster. So, for example, in 1794, George 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 \(\pi\). The Indian mathematician Srinivasa Ramanujan (1887-1920) discovered many new formulas for \(\pi \). In 1910, he obtained a formula for calculating \(\pi \) through the expansion of the arc tangent in a Taylor series:

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

With 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 period of time. In 1949, using ENIAC, a group of scientists led by John von Neumann (1903-1957) obtained 2037 decimal places of \(\pi \) in just 70 hours. David and Gregory Chudnovsky in 1987 obtained a formula with which they were able to set several records in the calculation \(\pi \):

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

Each member of the series gives 14 digits. In 1989, 1,011,196,691 decimal places were received. This formula is well suited for calculating \(\pi \) on personal computers. At the moment, 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 Pluff. It allows you to extract any hexadecimal digit of the number \(\pi \) without calculating the previous ones. The formula is called the "Bailey-Borwain-Pluff 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 computing \(\pi \). 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 four-core AMD Opteron processors, which provided a 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 set without the use of a supercomputer. For high performance, Fabrice used the formula of the Chudnovsky brothers. In total, the calculation took 131 days (103 days of calculation and 13 days of verification). Bellar's achievement showed that for such calculations it is not necessary to have a supercomputer.

Just six months later, François' record was broken by engineers Alexander Yi and Singer Kondo. To set a record of 5 trillion decimal places \(\pi \), 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 that had set their previous record and took a total of 371 days. At the end of 2013, Alexander and Singeru improved the record to 12.1 trillion digits of the number \(\pi \), which took them only 94 days to calculate. This improvement in performance 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 Yi and Singeru Kondo, which is 12.1 trillion decimal places of \(\pi \).

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

List of sources

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

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January 13, 2017

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What is common between a wheel from Lada Priora, a wedding ring and a saucer of your cat? Of course, you will say beauty and style, but I dare to argue with you. Pi! This is a number that unites all circles, circles and roundness, which include, in particular, my mother's ring, and the wheel from my father's favorite car, and even the saucer of my beloved cat Murzik. I'm willing to bet that in the ranking of the most popular physical and mathematical constants, the number Pi will undoubtedly take the first line. But what is behind it? Maybe some terrible curses of mathematicians? Let's try to understand this issue.

What is the number "Pi" and where did it come from?

Modern number notation π (Pi) appeared thanks to the English mathematician Johnson in 1706. This is the first letter of the Greek word περιφέρεια (periphery, or circumference). For those who have gone through mathematics for a long time, and besides, past, we recall that the number Pi is the ratio of the circumference of a circle to its diameter. The value is a constant, that is, it is constant for any circle, regardless of its radius. People have known about this since ancient times. So in ancient Egypt, the number Pi was taken equal to the ratio 256/81, and in the Vedic texts the value 339/108 is given, while Archimedes suggested the ratio 22/7. But neither these nor many other ways of expressing the number pi gave an accurate result.

It turned out that the number Pi is transcendental, respectively, and irrational. This means that it cannot be represented as a simple fraction. If it is expressed in terms of decimal, then the sequence of digits after the decimal point will rush to infinity, moreover, without periodically repeating. What does all of this mean? Very simple. Do you want to know the phone number of the girl you like? It can certainly be found in the sequence of digits after the decimal point of Pi.

Phone can be viewed here ↓

Pi number up to 10000 characters.

π= 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989..

Didn't find it? Then look.

In general, it can be not only a phone number, but any information encoded using numbers. For example, if we represent all the works of Alexander Sergeevich Pushkin in digital form, then they were stored in the number Pi even before he wrote them, even before he was born. In principle, they are still stored there. By the way, curses of mathematicians in π are also present, and not only mathematicians. In a word, Pi has everything, even thoughts that will visit your bright head tomorrow, the day after tomorrow, in a year, or maybe in two. This is very hard to believe, but even if we pretend to believe it, it will be even more difficult to get information from there and decipher it. So instead of delving into these numbers, it might be easier to approach the girl you like and ask her for a number? .. But for those who are not looking for easy ways, well, or just interested in what the number Pi is, I offer several ways to calculations. Count on health.

What is the value of Pi? Methods for its calculation:

1. Experimental method. If pi is the ratio of a circle's circumference to its diameter, then perhaps the first and most obvious way to find our mysterious constant would be to manually take all measurements and calculate pi using the formula π=l/d. Where l is the circumference of the circle and d is its diameter. Everything is very simple, you just need to arm yourself with a thread to determine the circumference, a ruler to find the diameter, and, in fact, the length of the thread itself, and a calculator if you have problems with division into a column. A saucepan or a jar of cucumbers can act as a measured sample, it doesn’t matter, the main thing? so that the base is a circle.

The considered calculation method is the simplest, but, unfortunately, it has two significant drawbacks that affect the accuracy of the resulting Pi number. Firstly, the error of measuring instruments (in our case, this is a ruler with a thread), and secondly, there is no guarantee that the circle we measure will have the correct shape. Therefore, it is not surprising that mathematics has given us many other methods for calculating π, where there is no need to make accurate measurements.

2. Leibniz series. There are several infinite series that allow you to accurately calculate the number of pi to a large number of decimal places. One of the simplest series is the Leibniz series. π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...
It's simple: we take fractions with 4 in the numerator (this is the one on top) and one number from the sequence of odd numbers in the denominator (this is the one on the bottom), sequentially add and subtract them with each other and get the number Pi. The more iterations or repetitions of our simple actions, the more accurate the result. Simple, but not effective, by the way, it takes 500,000 iterations to get the exact value of Pi to ten decimal places. That is, we will have to divide the unfortunate four as many as 500,000 times, and in addition to this, we will have to subtract and add the results obtained 500,000 times. Want to try?

3. The Nilakanta series. No time fiddling around with Leibniz next? There is an alternative. The Nilakanta series, although it is a bit more complicated, allows us to get the desired result faster. π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11 *12) - (4/(12*13*14) ... I think if you carefully look at the given initial fragment of the series, everything becomes clear, and comments are superfluous. On this we go further.

4. Monte Carlo method A rather interesting method for calculating pi is the Monte Carlo method. Such an extravagant name he got in honor of the city of the same name in the kingdom of Monaco. And the reason for this is random. No, it was not named by chance, it's just that the method is based on random numbers, and what could be more random than the numbers that fall on the Monte Carlo casino roulettes? The calculation of pi is not the only application of this method, as in the fifties it was used in the calculations of the hydrogen bomb. But let's not digress.

Let's take a square with a side equal to 2r, and inscribe in it a circle with a radius r. Now if you randomly put dots in a square, then the probability P that a point fits into a circle is the ratio of the areas of the circle and the square. P \u003d S cr / S q \u003d 2πr 2 / (2r) 2 \u003d π / 4.

Now from here we express the number Pi π=4P. It remains only to obtain experimental data and find the probability P as the ratio of hits in the circle N cr to hit the square N sq.. In general, the calculation formula will look like this: π=4N cr / N sq.

I would like to note that in order to implement this method, it is not necessary to go to the casino, it is enough to use any more or less decent programming language. Well, the accuracy of the results will depend on the number of points set, respectively, the more, the more accurate. I wish you good luck 😉

Tau number (instead of conclusion).

People who are far from mathematics most likely do not know, but it so happened that the number Pi has a brother who is twice as large as it. This is the number Tau(τ), and if Pi is the ratio of the circumference to the diameter, then Tau is the ratio of that length to the radius. And today there are proposals by some mathematicians to abandon the number Pi and replace it with Tau, since this is in many ways more convenient. But so far these are only proposals, and as Lev Davidovich Landau said: "A new theory begins to dominate when the supporters of the old one die out."