As the number pi is often called. What does Pi hide?

Today is the birthday of Pi, which, on the initiative of American mathematicians, is celebrated on March 14 at 1 hour and 59 minutes in the afternoon. This is connected with a more precise value of Pi: we are all accustomed to considering this constant as 3.14, but the number can be continued as follows: 3, 14159... Translating this into a calendar date, we get 03.14, 1:59.

Photo: AiF/ Nadezhda Uvarova

Professor of the Department of Mathematical and Functional Analysis of South Ural State University Vladimir Zalyapin says that July 22 should still be considered “Pi day”, because in the European date format this day is written as 22/7, and the value of this fraction is approximately equal to the value of Pi .

“The history of the number that gives the ratio of the circumference to the diameter of the circle goes back to ancient times,” says Zalyapin. - Already the Sumerians and Babylonians knew that this ratio does not depend on the diameter of the circle and is constant. One of the first mentions of the number Pi can be found in the texts Egyptian scribe Ahmes(circa 1650 BC). The ancient Greeks, who borrowed a lot from the Egyptians, contributed to the development of this mysterious quantity. According to the legend, Archimedes was so carried away by calculations that he did not notice how Roman soldiers took his hometown of Syracuse. When the Roman soldier approached him, Archimedes shouted in Greek: “Don’t touch my circles!” In response, the soldier stabbed him with a sword.

Plato received a fairly accurate value of Pi for his time - 3.146. Ludolf van Zeilen spent most of his life calculating the first 36 decimal places of Pi, and they were engraved on his tombstone after his death."

Irrational and abnormal

According to the professor, at all times the pursuit of calculating new decimal places was determined by the desire to obtain the exact value of this number. It was assumed that Pi was rational and could therefore be expressed as a simple fraction. And this is fundamentally wrong!

The number Pi is also popular because it is mystical. Since ancient times, there has been a religion of worshipers of the constant. In addition to the traditional value of Pi - a mathematical constant (3.1415...), expressing the ratio of the circumference of a circle to its diameter, there are many other meanings of the number. Such facts are interesting. In the process of measuring the dimensions of the Great Pyramid of Giza, it turned out that it has the same ratio of height to the perimeter of its base as the radius of a circle to its length, that is, ½ Pi.

If you calculate the length of the Earth's equator using Pi to the ninth decimal place, the error in the calculations will be only about 6 mm. Thirty-nine decimal places in Pi are enough to calculate the circumference of the circle surrounding known cosmic objects in the Universe, with an error no greater than the radius of a hydrogen atom!

The study of Pi also includes mathematical analysis. Photo: AiF/ Nadezhda Uvarova

Chaos in numbers

According to a mathematics professor, in 1767 Lambert established the irrationality of the number Pi, that is, the impossibility of representing it as a ratio of two integers. This means that the sequence of decimal places of Pi is chaos embodied in numbers. In other words, the “tail” of decimal places contains any number, any sequence of numbers, any texts that were, are and will be, but it’s just not possible to extract this information!

“It is impossible to know the exact value of Pi,” continues Vladimir Ilyich. - But these attempts are not abandoned. In 1991 Chudnovsky achieved a new 2260000000 decimal places of the constant, and in 1994 - 4044000000. After that, the number of correct digits of Pi increased like an avalanche.”

Chinese holds world record for memorizing Pi Liu Chao, who was able to remember 67,890 decimal places without error and reproduce them within 24 hours and 4 minutes.

About the “golden ratio”

By the way, the connection between “pi” and another amazing quantity - the golden ratio - has never actually been proven. People have long noticed that the “golden” proportion - also known as the number Phi - and the number Pi divided by two differ from each other by less than 3% (1.61803398... and 1.57079632...). However, for mathematics, these three percent are too significant a difference to consider these values ​​identical. In the same way, we can say that the Pi number and the Phi number are relatives of another well-known constant - the Euler number, since the root of it is close to half the Pi number. One half of Pi is 1.5708, Phi is 1.6180, the root of E is 1.6487.

This is only part of the value of Pi. Photo: Screenshot

Pi's birthday

At South Ural State University, the birthday of the constant is celebrated by all teachers and mathematics students. This has always been the case - it cannot be said that interest has appeared only in recent years. The number 3.14 is even welcomed with a special holiday concert!

There is no cyclicity or system in the numbers after the decimal point, that is, in the decimal expansion of Pi there is any sequence of numbers that you can imagine (including a very rare sequence in mathematics of a million non-trivial zeros, predicted by the German mathematician Bernhardt Riemann back in 1859).

This means that Pi, in encoded form, contains all written and unwritten books, and in general any information that exists (which is why the calculations of the Japanese professor Yasumasa Kanada, who recently determined the number Pi to 12411 trillion decimal places, were immediately classified - with such a volume of data it is not difficult to reconstruct the contents of any secret document printed before 1956, although this data is not enough to determine the location of any person, this requires at least 236,734 trillion decimal places - it is assumed that such work is now being carried out in Pentagon (using quantum computers, the clock speed of which is already approaching sound speed).

Any other constant can be defined through the number Pi, including the fine structure constant (alpha), the golden proportion constant (f=1.618...), not to mention the number e - this is why the number pi is found not only in geometry, but also in the theory of relativity , quantum mechanics, nuclear physics, etc. Moreover, scientists have recently found that it is through Pi that it is possible to determine the location of elementary particles in the Table of Elementary Particles (previously they tried to do this through Woody’s Table), and the message that in the recently deciphered human DNA, the number Pi is responsible for the structure of DNA itself (enough complex, it should be noted), produced the effect of a bomb exploding!

According to Dr. Charles Cantor, under whose leadership DNA was deciphered: “It seems that we have come to the solution to some fundamental problem that the universe has thrown at us. The number Pi is everywhere, it controls all processes known to us, while remaining unchanged! Who controls the number Pi itself? No answer yet.” In fact, Cantor is disingenuous, there is an answer, it’s just so incredible that scientists prefer not to make it public, fearing for their own lives (more on that later): the number Pi controls itself, it’s reasonable! Nonsense? Do not hurry.

After all, Fonvizin also said that “in human ignorance, it is very comforting to consider everything as nonsense that you don’t know.

Firstly, conjectures about the reasonableness of numbers in general have long been visited by many famous mathematicians of our time. Norwegian mathematician Niels Henrik Abel wrote to his mother in February 1829: “I have received confirmation that one of the numbers is reasonable. I spoke to him! But what scares me is that I can't figure out what this number is. But maybe this is for the better. The Number warned me that I would be punished if It was revealed.” Who knows, Nils would have revealed the meaning of the number that spoke to him, but on March 6, 1829, he passed away.

1955, Japanese Yutaka Taniyama puts forward the hypothesis that “each elliptic curve corresponds to a certain modular form” (as is known, on the basis of this hypothesis Fermat’s theorem was proven). On September 15, 1955, at the international mathematical symposium in Tokyo, where Taniyama announced his hypothesis, in response to a journalist’s question: “How did you come up with this?” - Taniyama replies: “I didn’t think of it, the number told me about it over the phone.”

The journalist, thinking that this was a joke, decided to “support” her: “Did it tell you the phone number?” To which Taniyama seriously replied: “It seems that this number has been known to me for a long time, but I can now report it only after three years, 51 days, 15 hours and 30 minutes.” In November 1958, Taniyama committed suicide. Three years, 51 days, 15 hours and 30 minutes is 3.1415. Coincidence? May be. But here's another one, even stranger. The Italian mathematician Sella Quitino also spent several years, as he vaguely put it, “keeping in touch with one cute number.” The figure, according to Quitino, who was already in a psychiatric hospital at that time, “promised to say his name on his birthday.” Could Quitino have lost his mind so much as to call the number Pi a number, or was he deliberately confusing the doctors? It is not clear, but on March 14, 1827, Quitino passed away.

And the most mysterious story is connected with the “great Hardy” (as you all know, this is what contemporaries called the great English mathematician Godfrey Harold Hardy), who, together with his friend John Littlewood, is famous for his work in number theory (especially in the field of Diophantine approximations) and function theory ( where friends became famous for their study of inequalities). As you know, Hardy was officially unmarried, although he repeatedly stated that he was “engaged to the queen of our world.” Fellow scientists more than once heard him talking to someone in his office; no one had ever seen his interlocutor, although his voice - metallic and slightly creaky - had long been the talk of the town at Oxford University, where he worked in recent years . In November 1947, these conversations stop, and on December 1, 1947, Hardy is found in a city dump, with a bullet in his stomach. The version of suicide was also confirmed by a note in which Hardy’s hand wrote: “John, you stole the queen from me, I don’t blame you, but I can no longer live without her.”

Is this story related to the number Pi? It’s still unclear, but isn’t it interesting?+

Is this story related to the number Pi? It’s still unclear, but isn’t it interesting?
Generally speaking, you can collect a lot of similar stories, and, of course, not all of them are tragic.
But, let's move on to “secondly”: how can a number even be reasonable? Yes, very simple. The human brain contains 100 billion neurons, the number of decimal places of Pi tends to infinity, in general, according to formal criteria, it can be reasonable. But if you believe the work of the American physicist David Bailey and Canadian mathematicians Peter

Borwin and Simon Ploofe, the sequence of decimal places in Pi is subject to chaos theory; roughly speaking, the number Pi is chaos in its original form. Can chaos be intelligent? Certainly! Just like a vacuum, despite its apparent emptiness, as is known, it is by no means empty.

Moreover, if you wish, you can represent this chaos graphically - to make sure that it can be reasonable. In 1965, an American mathematician of Polish origin Stanislaw M. Ulam (he was the one who came up with the key idea for the design of a thermonuclear bomb), while attending one very long and very boring (in his words) meeting, in order to somehow have fun, began to write numbers on checkered paper , included in the number Pi.

Putting 3 in the center and moving counterclockwise in a spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Without any second thought, he simultaneously circled all the prime numbers with black circles. Soon, to his surprise, the circles with amazing tenacity began to line up along straight lines - what happened was very similar to something reasonable. Especially after Ulam generated a color picture based on this drawing using a special algorithm.

Actually, this picture, which can be compared with both a brain and a stellar nebula, can safely be called the “brain of Pi.” Approximately with the help of such a structure, this number (the only reasonable number in the universe) controls our world. But how does this control take place? As a rule, with the help of the unwritten laws of physics, chemistry, physiology, astronomy, which are controlled and adjusted by a reasonable number. The above examples show that the intelligent number is also deliberately personified, communicating with scientists as a kind of superpersonality. But if so, did the number Pi come to our world in the guise of an ordinary person?

Complex issue. Maybe it came, maybe it didn’t, there is no reliable method for determining this and there cannot be, but if this number is determined by itself in all cases, then we can assume that it came into our world as a person on the day corresponding to its meaning. Of course, the ideal date of Pi’s birth is March 14, 1592 (3.141592), however, unfortunately, there are no reliable statistics for this year - we only know that it was in this year, on March 14, that George Villiers Buckingham, the Duke of Buckingham from “ The Three Musketeers." He was an excellent fencer, knew a lot about horses and falconry - but was he Pi? Hardly. Duncan MacLeod, born on March 14, 1592, in the mountains of Scotland, could ideally lay claim to the role of the human embodiment of the number Pi - if he were a real person.

But the year (1592) can be determined according to its own, more logical calendar for Pi. If we accept this assumption, then there are many more candidates for the role of Pi.+

The most obvious of them is Albert Einstein, born March 14, 1879. But 1879 is 1592 relative to 287 BC! Why exactly 287? Yes, because it was in this year that Archimedes was born, who for the first time in the world calculated the number Pi as the ratio of the circumference to the diameter and proved that it is the same for any circle!

Coincidence? But aren’t there a lot of coincidences, don’t you think?

In what personality Pi is personified today is not clear, but in order to see the meaning of this number for our world, you don’t need to be a mathematician: Pi manifests itself in everything that surrounds us. And this, by the way, is very typical for any intelligent being, which, without a doubt, is Pi!

NUMBER p – the ratio of the circumference of a circle to its diameter, is a constant value 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 notation came into use with the work of Leonhard Euler in 1736, but 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 related to circles and round bodies forced us to look for 241 approximations using rational numbers already in ancient times. Information that the circle is exactly three times longer than the diameter is found in the cuneiform tablets of Ancient Mesopotamia. Same number value p is also in the text of the Bible: “And he made a sea cast of copper, ten cubits from one end to the other, completely round, five cubits high, and a string of thirty cubits encircled it” (1 Kings 7:23). The ancient Chinese believed the same. But already in 2 thousand BC. the ancient Egyptians used a more precise value for the number 241, which is obtained from the formula for the area of ​​a circle's diameter d:

This rule from the 50th problem of the Rhind papyrus corresponds to the value 4(8/9) 2 » 3.1605. The Rhind 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 has only been established that the text was created in the second half of the 19th century. BC. Although how the Egyptians received the formula itself is unclear 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, around 1900 BC, there is another interesting problem about calculating the surface of a basket "with a 4½ hole". 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.

To understand how ancient scientists obtained this or that result, you need to try to solve the problem using only the knowledge and calculation techniques of that time. This is exactly what researchers of ancient texts do, but the solutions they manage to find are not necessarily “the same.” Very often, several solution options are offered for one problem; everyone can choose to their liking, but no one can claim that this was the solution that was used in ancient times. 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 is the 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: to a first approximation, the area of ​​a circle S equal to the difference between the area of ​​a square and its side d and the total area of ​​four small squares A with the side d:

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

From the 6th century 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– radius of the circle, l – its length), and the area of ​​the circle is equal to half the product of the circumference and radius:

S = ½ l R = p R 2 .

These proofs are attributed to Eudoxus of Cnidus and Archimedes.

In the 3rd century. BC. Archimedes in his essay About measuring a circle calculated the perimeters of regular polygons inscribed in a circle and circumscribed around it (Fig. 2) - from a 6- to a 96-gon. Thus he established that the number p is 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, creator of trigonometry Claudius Ptolemy (2nd century), but it did not come into use.

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

The search for a more accurate approximation p continued in the future. For example, al-Kashi (first half of the 15th century) in Treatise on the Circle(1427) calculated 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 Ludolf Van Zeijlen (1540–1610) found 32 correct decimal places for it (published posthumously in 1615), an approximation called the Ludolf number.

Number p appears not only when solving geometric problems. Since the time of F. Vieta (1540–1603), the search for the limits of certain arithmetic sequences compiled according to simple laws led to the same number p. In this regard, in determining the number p Almost all famous mathematicians took part: F. Viet, H. Huygens, J. Wallis, G. W. Leibniz, L. Euler. They received various expressions for 241 in the form of an infinite product, a 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 unknown how he arrived at this result.

In 1665 John Wallis (1616–1703) proved that

This formula bears his name. It is of little use for the practical determination of the number 241, but is useful in various theoretical discussions. It went down in the history of science as one of the first examples of endless works.

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

expressing a 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 calculations 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 signs.

Using the same series for arctg x and formulas

number value p was obtained on a computer with an accuracy of one hundred thousand decimal places. This kind of calculation is of interest in connection with the concept of random and pseudorandom numbers. Statistical processing of an ordered collection 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 numbers p more accurate than just 3.14. For example, having learned the following quatrain, you can easily name seven decimal places p:

You just have 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 saying was proposed by Ya.I. Perelman.

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

“Learn and know the number behind the number, how to notice luck” ( p"3.14159265359).

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

This is what 101 numbers look like p no 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 meaning of a number p calculated with millions of correct digits, but such precision is not needed in any calculation. But the possibility of analytically determining 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- an irrational number, but only in 1766 the German mathematician Johann Heinrich Lambert (1728–1777), based on the relationship between 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 Louise 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 xn– 1 + … + a 1 x+a 0 = 0 with integer coefficients. This proof put an end to the history of the ancient mathematical problem of squaring the circle. For millennia, this problem defied the efforts of mathematicians; the expression “squaring the circle” became synonymous with an unsolvable problem. And the whole point turned out to be 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 at the University of Munich. On the pedestal under his name there is a circle intersected by a square of equal area, inside which the letter is inscribed p.

Marina Fedosova

Pi" is a transcendental number, or in simple words it cannot be the root of some polynomial with integer coefficients. It can be designated as a real number or as an indirect number that is not algebraic.

The number "Pi" is 3.1415926535 8979323846 2643383279 5028841971 6939937510...

Pi" can be correlated with the fractional number 22/7, the so-called “triple octave” symbol. The ancient Greek priests knew this number. In addition, even ordinary residents could use it to solve any everyday problems, and also use it to design such complex structures as tombs.
According to scientist and researcher Hayens, a similar number can be traced among the ruins of Stonehenge, and also found in the Mexican pyramids.

Pi" Ahmes, a famous engineer at that time, mentioned in his writings. He tried to calculate it as accurately as possible by measuring the diameter of the circle using the squares drawn inside it. Probably in some sense this number has some mystical, sacred meaning for the ancients.

Pi" is essentially the most mysterious mathematical symbol. It can be classified as delta, omega, etc. It represents a relationship that will turn out to be exactly the same, regardless of where the observer will be in the universe. In addition, it will be unchanged from the object of measurement.

Most likely, the first person who decided to calculate the number "Pi" using a mathematical method is Archimedes. He decided to draw regular polygons in a circle. Considering the diameter of a circle to be one, the scientist designated the perimeter of a polygon drawn in a circle, considering the perimeter of an inscribed polygon as an upper estimate, and as a lower estimate of the circumference

What is the number "Pi"

***

What do a Lada Priora wheel, a wedding ring and your cat's saucer have in common? 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 in particular include my mother’s ring, the wheel from my father’s favorite car, and even the saucer of my favorite cat Murzik. I'm willing to bet that in the ranking of the most popular physical and mathematical constants, Pi will undoubtedly take first place. But what is hidden behind it? Maybe some terrible curse words from mathematicians? Let's try to understand this issue.

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

Modern number designation π (Pi) appeared thanks to the English mathematician Johnson in 1706. This is the first letter of the Greek word περιφέρεια (periphery, or circle). For those who took mathematics a long time ago, and besides, by no means, let us remind you that the number Pi is the ratio of the circumference of a circle to its diameter. The value is a constant, that is, constant for any circle, regardless of its radius. People knew about this in ancient times. Thus, in ancient Egypt, the number Pi was taken to be equal to the ratio 256/81, and in Vedic texts the value is given as 339/108, while Archimedes proposed 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 and, accordingly, irrational. This means that it cannot be represented as a simple fraction. If we express it in decimal terms, then the sequence of digits after the decimal point will rush to infinity, and, moreover, without periodically repeating itself. What does all of this mean? Very simple. Do you want to know the phone number of the girl you like? It can probably be found in the sequence of digits after the decimal point of Pi.

You can see the phone number here ↓

Pi number accurate to 10,000 digits.

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 take a look.

In general, this can be not only a phone number, but any information encoded using numbers. For example, if you imagine 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, the curses of mathematicians in π are also present, and not only mathematicians. In a word, the number Pi contains everything, even thoughts that will visit your bright head tomorrow, the day after tomorrow, in a year, or maybe in two. This is very difficult to believe, but even if we imagine that we believe it, it will be even more difficult to obtain information from it and decipher it. So, instead of delving into these numbers, maybe it’s easier to approach the girl you like and ask her number?.. But for those who are not looking for easy ways, or simply interested in what the number Pi is, I offer several ways calculations. Consider it healthy.

What is Pi equal to? Methods for calculating it:

1. Experimental method. If the number Pi is the ratio of the circumference of a circle to its diameter, then the first, perhaps the most obvious way to find our mysterious constant will be to manually make all the measurements and calculate the number 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 long division. The role of the sample to be measured can be a saucepan or a jar of cucumbers, it doesn’t matter, the main thing is? so that there is a circle at the base.

The considered method of calculation is the simplest, but, unfortunately, it has two significant drawbacks that affect the accuracy of the resulting Pi number. Firstly, the error of the measuring instruments (in our case, a ruler with a thread), and secondly, there is no guarantee that the circle we are measuring 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 precise measurements.

2. Leibniz series. There are several infinite series that allow you to accurately calculate 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 what’s on top) and one number from the sequence of odd numbers in the denominator (this is what’s below), 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 obtained results 500,000 times. Want to try?

3. Nilakanta series. Don't have time to tinker with the Leibniz series? There is an alternative. The Nilakanta series, although it is a little more complicated, allows us to quickly get the desired result. 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 look carefully at the given initial fragment of the series, everything becomes clear, and comments are unnecessary. Let's move on with this.

4. Monte Carlo method A rather interesting method for calculating Pi is the Monte Carlo method. It got such an extravagant name in honor of the city of the same name in the kingdom of Monaco. And the reason for this is coincidence. No, it was not named by chance, the method is simply based on random numbers, and what could be more random than the numbers that appear on the roulette tables of the Monte Carlo casino? Calculating Pi is not the only application of this method; in the fifties it was used in calculations of the hydrogen bomb. But let's not get distracted.

Take a square with a side equal to 2r, and inscribe a circle with radius r. Now if you put dots in a square at random, then the probability P The fact that a point falls into a circle is the ratio of the areas of the circle and the square. P=S kr /S kv =2πr 2 /(2r) 2 =π/4.

Now let's express the number Pi from here π=4P. All that remains is to obtain experimental data and find the probability P as the ratio of hits in the circle N cr to hitting the square N sq.. In general, the calculation formula will look like this: π=4N cr / N square.

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

Tau number (Instead of a conclusion).

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