What is "pi" is known to absolutely everyone. But the number familiar to everyone from school appears in many situations that have nothing to do with circles. It can be found in probability theory, in the Stirling formula for calculating the factorial, in solving problems with complex numbers, and in other unexpected and far from geometry areas of mathematics. The English mathematician August de Morgan once called "pi" "... the mysterious number 3.14159... that climbs through the door, through the window and through the roof."
This mysterious number, associated with one of the three classic problems of Antiquity - the construction of a square whose area is equal to the area of a given circle - entails a trail of dramatic historical and curious entertaining facts.
- Some interesting facts about pi
- 1. Did you know that the first person to use the symbol "pi" for the number 3.14 was William Jones from Wales, and this happened in 1706.
- 2. Did you know that the world record for memorizing the number Pi was set on June 17, 2009 by the Ukrainian neurosurgeon, Doctor of Medical Sciences, Professor Andrey Slyusarchuk, who kept 30 million of its signs in memory (20 volumes of text).
- 3. Did you know that in 1996 Mike Keith wrote a short story called "Cadeic Cadenze", in his text the length of the words corresponded to the first 3834 digits of pi.
It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who drew attention to the fact that on March 14 (in the American spelling - 3.14) exactly at 01:59 the date and time will coincide with the first digits of Pi = 3.14159.
March 14, 1879 was also the birthday of the creator of the theory of relativity, Albert Einstein, which makes this day even more attractive for all lovers of mathematics.
In addition, mathematicians also celebrate the day of the approximate value of Pi, which falls on July 22 (22/7 in the European date format).
"At this time, they read laudatory speeches in honor of the number Pi and its role in the life of mankind, draw dystopian pictures of the world without Pi, eat pies with the image of the Greek letter Pi or with the first digits of the number itself, solve mathematical puzzles and riddles, and also dance" , writes Wikipedia.
Numerically, pi starts as 3.141592 and has an infinite mathematical duration.
French scientist Fabrice Bellard calculated the number Pi with record accuracy. This is reported on his official website. The latest record is about 2.7 trillion (2 trillion 699 billion 999 million 990 thousand) decimal places. The previous achievement belongs to the Japanese, who calculated the constant with an accuracy of 2.6 trillion decimal places.
It took Bellar about 103 days to calculate. All calculations were carried out on a home computer, the cost of which lies within 2000 euros. For comparison, the previous record was set on the T2K Tsukuba System supercomputer, which took about 73 hours to run.
Initially, the Pi number appeared as the ratio of the circumference of a circle to its diameter, so its approximate value was calculated as the ratio of the perimeter of a polygon inscribed in a circle to the diameter of this circle. Later, more advanced methods appeared. Pi is currently calculated using rapidly convergent series, such as those proposed by Srinivas Ramanujan in the early 20th century.
Pi was first calculated in binary and then converted to decimal. This was done in 13 days. A total of 1.1 terabytes of disk space is required to store all the numbers.
Such calculations have not only applied value. So, now there are many unsolved problems associated with Pi. The question of the normality of this number has not been resolved. For example, it is known that pi and e (the base of the exponent) are transcendental numbers, that is, they are not the roots of any polynomial with integer coefficients. In this case, however, whether the sum of these two fundamental constants is a transcendental number or not is still unknown.
Moreover, it is still not known whether all the digits from 0 to 9 occur in the decimal notation of pi an infinite number of times.
In this case, the ultra-precise calculation of a number is a convenient experiment, the results of which allow us to formulate hypotheses regarding certain features of the number.
The number is calculated according to certain rules, and in any calculation, in any place and at any time, at a certain place in the record of the number is the same digit. This means that there is a certain law according to which a certain figure is put in a number in a certain place. Of course, this law is not simple, but the law still exists. And, therefore, the numbers in the record of the number are not random, but regular.
Pi is counted: PI = 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... - 4/n + 4/(n+2)
Search for Pi or division by a column:
Pairs of integers that, when divided, give a large approximation to the number Pi. The division was done by a "column" to get around the limitations on the length of Visual Basic 6 floating point numbers.
Pi = 3.14159265358979323846264>33832795028841 971...
Exotic methods for calculating pi, such as using the theory of probability or prime numbers, also include the method invented by G.A. Galperin, and called Pi Billiard, which is based on the original model. When two balls collide, the smaller of which is between the larger one and the wall, and the larger one moves towards the wall, the number of collisions of the balls makes it possible to calculate Pi with an arbitrarily large predetermined accuracy. You just need to start the process (you can also use it on a computer) and count the number of hits of the balls. The software implementation of this model is not yet known.
In every book on entertaining mathematics, you will certainly find a history of calculating and refining the value of the number "pi". At first, in ancient China, Egypt, Babylon and Greece, fractions were used for calculations, for example, 22/7 or 49/16. In the Middle Ages and the Renaissance, European, Indian and Arabic mathematicians refined the value of "pi" to 40 decimal places, and by the beginning of the Computer Age, the number of characters was increased to 500 by the efforts of many enthusiasts. Such accuracy is of purely scientific interest (more on that below) , for practice, 11 signs after the dot are enough within the Earth.
Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that we, having discarded the twelfth digit "pi" after the point when calculating the length of the meridian, will be mistaken by several millimeters. And when calculating the length of the Earth's orbit during rotation around the Sun (as you know, R = 150 * 106 km = 1.5 * 1014 mm), for the same accuracy, it is enough to use "pi" with fourteen digits after the point. The average distance from the Sun to Pluto, the most distant planet in the solar system, is 40 times the average distance from the Earth to the Sun.
To calculate the length of Pluto's orbit with an error of a few millimeters, sixteen "pi" signs are enough. Yes, what is there to trifle - the diameter of our Galaxy is about 100,000 light years (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm., And back in the 27th century, 34 pi signs were obtained, redundant for such distances.
What is the complexity of calculating the value of "pi"? The fact is that it is not only irrational (that is, it cannot be expressed as a fraction P / Q, where P and Q are integers), but it cannot yet be the root of an algebraic equation. A number, for example, an irrational one, cannot be represented by a ratio of integers, but it is the root of the equation X2-2=0, and for the numbers "pi" and e (Euler's constant), such an algebraic (non-differential) equation cannot be specified. Such numbers (transcendental) are calculated by considering a process and are refined by increasing the steps of the process under consideration. The most “simple” way is to inscribe a regular polygon in a circle and calculate the ratio of the perimeter of the polygon to its “radius”...pages marsu
Number explains the world
It seems that two American mathematicians have managed to get closer to unraveling the mystery of the number pi, which in purely mathematical terms represents the ratio of the circumference of a circle to its diameter, reports Der Spiegel.
As an irrational value, it cannot be represented as a complete fraction, so an endless series of numbers follows the decimal point. This property has always attracted mathematicians who sought to find, on the one hand, a more accurate value of pi, and, on the other hand, its generalized formula.
However, mathematicians David Bailey of the Lawrence Berkeley National Laboratory in California and Richard Grendel of Reed College in Portland looked at the number from a different angle - they tried to find some meaning in the seemingly chaotic series of numbers after the decimal point. As a result, it was found that combinations of the following numbers are regularly repeated - 59345 and 78952.
But so far they cannot answer the question of whether the repetition is random or regular. The question of the pattern of repetition of certain combinations of numbers, and not only in the number pi, is one of the most difficult in mathematics. But now we can say something more definite about this number. The discovery paves the way for unraveling the number pi and, in general, for determining its essence - whether it is normal for our world or not.
Both mathematicians have been interested in the number pi since 1996, and since that time they have had to abandon the so-called "number theory" and pay attention to the "chaos theory", which is now their main weapon. Researchers construct based on the display of the number pi - its most common form is 3.14159 ... - series of numbers between zero and one - 0.314, 0.141, 0.415, 0.159 and so on. Therefore, if the number pi is indeed chaotic, then the series of numbers starting from zero must also be chaotic. But there is no answer to this question yet. To unravel the secret of pi, like its older brother - the number 42, with the help of which many researchers are trying to explain the secret of the universe, has yet to be."
Interesting data about the distribution of pi digits.
(Programming is the greatest achievement of mankind. Thanks to it, we regularly learn what we don’t need to know at all, but it’s very interesting)
Calculated (for a million decimal places):
zeros = 99959,
units = 99758,
twos = 100026,
triplets = 100229,
fours = 100230,
fives = 100359,
sixes = 99548,
sevens = 99800,
eights = 99985,
nines = 100106.
In the first 200,000,000,000 decimal places of pi, digits occurred with the following frequency:
"0" : 20000030841;
"1" : 19999914711;
"2" : 20000136978;
"3" : 20000069393
"4" : 19999921691;
"5" : 19999917053;
"6" : 19999881515;
"7" : 19999967594
"8" : 20000291044;
"9" : 19999869180;
That is, the numbers are distributed almost evenly. Why? Because according to modern mathematical concepts, with an infinite number of digits, they will be exactly equal, in addition, there will be as many ones as twos and triples combined, and even as many as all the other nine digits combined. But here to know where to stop, to seize the moment, so to speak, where they are really evenly divided.
And yet - in the digits of Pi, you can expect the appearance of any predetermined sequence of digits. For example, the most common arrangements were found in the following numbers in a row:
01234567891: from 26.852.899.245
01234567891: from 41,952,536,161
01234567891: from 99.972.955.571
01234567891: from 102,081,851,717
01234567891: from 171,257,652,369
01234567890: from 53,217,681,704
27182818284: c 45,111,908,393 are the digits of e. (
There was such a joke: scientists found the last number in the record of Pi - it turned out to be the number e, almost hit)
You can search in the first ten thousand characters of Pi for your phone number or date of birth, if it doesn’t work, then look in 100,000 characters.
In the number 1 / Pi, starting from 55,172,085,586 signs, there are 3333333333333, isn't it amazing?
In philosophy, the accidental and the necessary are usually contrasted. So the signs of pi are random? Or are they necessary? Let's say the third digit of pi is "4". And regardless of who would calculate this pi, in what place and at what time he would not do it, the third sign will necessarily always be equal to "4".
Relationship between pi, phi and the Fibonacci series. Relationship between the number 3.1415916 and the number 1.61803 and the Pisa sequence.
- More interesting:
- 1. In decimal positions of Pi, 7, 22, 113, 355 is the number 2. Fractions 22/7 and 355/113 are good approximations to Pi.
- 2. Kochansky found that Pi is the approximate root of the equation: 9x^4-240x^2+1492=0
- 3. If you write the capital letters of the English alphabet clockwise in a circle and cross out the letters that have symmetry from left to right: A, H, I, M, O, T, U, V, W, X, Y, then the remaining letters form groups according to 3,1,4,1,6 lit.
- (A) BCDEFG (HI) JKL (M) N (O) PQRS (TUVWXY) Z
- 6 3 1 4 1
- So the English alphabet must begin with the letter H, I or J, and not with the letter A :)
And what about us? And it follows from this that any conceived sequence of digits can be found in the decimal tail of pi. Your phone number? Please, and more than once (you can check here, but keep in mind that this page weighs about 300 megabytes, so you will have to wait for the download. You can download a miserable million characters here or take a word: any sequence of digits in decimal places of pi early or late there. Any!
For more exalted readers, another example can be offered: if you encrypt all the letters with numbers, then in the decimal expansion of the number pi you can find all the world literature and science, and the recipe for making bechamel sauce, and all the sacred books of all religions. I'm not kidding, this is hard scientific fact. After all, the sequence is INFINITE and the combinations are not repeated, therefore it contains ALL combinations of numbers, and this has already been proven. And if everything, then everything. Including those that correspond to the book you have chosen.
And this again means that it contains not only all the world literature that has already been written (in particular, those books that were burned, etc.), but also all the books that WILL be written.
It turns out that this number (the only reasonable number in the universe!) And governs our world.
The question is how to find them there...
And on this day, Albert Einstein was born, who predicted ... but why didn’t he predict! ...even dark energy.
This world was shrouded in deep darkness.
Let there be light! And here comes Newton.
But Satan did not wait long for revenge.
Einstein came - and everything was as before.
They correlate well - pi and Albert...
Theories arise, develop and...
Bottom line: Pi is not equal to 3.14159265358979....
This is a delusion based on the erroneous postulate of identifying the flat Euclidean space with the real space of the Universe.
Brief explanation of why pi is not generally equal to 3.14159265358979...
This phenomenon is associated with the curvature of space. The lines of force in the universe at considerable distances are not perfectly straight, but slightly curved lines. We have already matured to the moment of stating the fact that in the real world there are no perfectly straight lines, ideally flat circles, ideal Euclidean space. Therefore, we must imagine any circle of one radius on a sphere of much larger radius.
We are mistaken in thinking that space is flat, "cubic". The universe is not cubic, not cylindrical, much less pyramidal. The universe is spherical. The only case in which a plane can be ideal (in the sense of "non-curved") is when such a plane passes through the center of the universe.
Of course, the curvature of a CD-ROM can be neglected, since the diameter of a CD is much smaller than the diameter of the Earth, much less the diameter of the Universe. But one should not neglect the curvature in the orbits of comets and asteroids. The indestructible Ptolemaic belief that we are still at the center of the universe can cost us dearly.
Below are the axioms of a flat Euclidean ("cubic" Cartesian) space and an additional axiom formulated by me for a spherical space.
Axioms of flat consciousness:
through 1 point you can draw an infinite number of lines and an infinite number of planes.
through 2 points you can draw 1 and only 1 straight line through which you can draw an infinite number of planes.
through 3 points, in the general case, it is impossible to draw a single straight line and one, and only one, plane. Additional axiom for spherical consciousness:
through 4 points, in the general case, it is impossible to draw a single line, not a single plane, and one and only one sphere. Arsentiev Alexey Ivanovich
A bit of mysticism. PI number Is it reasonable?
Through the number Pi, any other constant can be defined, including the fine structure constant (alpha), the golden ratio constant (f=1.618...), not to mention the number e - that is why the number pi occurs not only in geometry, but also in theory of relativity, quantum mechanics, nuclear physics, etc. Moreover, scientists have recently found that it is through Pi that one can determine the location of elementary particles in the Table of elementary particles (previously they tried to do this through the Woody Table), and the message that in the recently deciphered human DNA, the Pi number is responsible for the DNA structure itself (enough complex, it should be noted), produced the effect of an exploding bomb!
According to Dr. Charles Cantor, under whose leadership DNA was deciphered: “It seems that we have come to the solution of some fundamental problem that the universe has thrown at us. The number Pi is everywhere, it controls all the processes known to us, while remaining unchanged! does it control Pi itself? There is no answer yet."
In fact, Kantor is cunning, 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): Pi controls itself, it is reasonable! Nonsense? Do not hurry. After all, even Fonvizin said that "in human ignorance it is very comforting to consider everything as nonsense that you do not know."
First, conjectures about the reasonableness of numbers in general have long visited many famous mathematicians of our time. Norwegian mathematician Nils Henrik Abel wrote to his mother in February 1829: “I received confirmation that one of the numbers is reasonable. I talked to him! But it scares me that I cannot determine what this number is. But maybe this is for the best. The Number warned me that I would be punished if It was revealed." Who knows, Niels would have revealed the meaning of the number that spoke to him, but on March 6, 1829, he died.
1955, Japanese Yutaka Taniyama puts forward the hypothesis that "every elliptic curve corresponds to a certain modular form" (as is known, Fermat's theorem was proved on the basis of this hypothesis). September 15, 1955, at the International Mathematical Symposium in Tokyo, where Taniyama announced his conjecture, to the question of a journalist: "How did you think of this?" - Taniyama replies: "I did not think of it, the number told me about it on the phone." The journalist, thinking that this was a joke, decided to "support" her: "Did it tell you the phone number?" To which Taniyama replied seriously: "It seems that this number has been known to me for a long time, but now I can tell 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 something even stranger. The Italian mathematician Sella Quitino also, for several years, as he himself vaguely put it, "kept in touch with one cute figure." The figure, according to Kvitino, who was already in a psychiatric hospital, "promised to tell her name on her birthday." Could Kvitino have lost his mind so much as to call the number Pi a number, or was he deliberately confusing doctors? It is not clear, but on March 14, 1827, Kvitino died.
And the most mysterious story is connected with the "great Hardy" (as you all know, 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 the study of inequalities). As you know, Hardy was officially unmarried, although he repeatedly stated that he was "betrothed to the queen of our world." Fellow scientists have heard him talking to someone in his office more than once, no one has ever seen his interlocutor, although his voice - metallic and slightly raspy - has 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 the city dump, with a bullet in his stomach. The version of suicide was also confirmed by a note, where Hardy's hand was written: "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 pi? It's not clear yet, but isn't it curious?
Generally speaking, one can dig up a lot of such stories, and, of course, not all of them are tragic.
But, let's move on to the "second": how can a number be reasonable at all? Yes, very simple. The human brain contains 100 billion neurons, the number of pi after the decimal point generally tends to infinity, in general, according to formal signs, it can be reasonable. But if you believe the work of the American physicist David Bailey and Canadian mathematicians Peter Borvin and Simon Ploof, the sequence of decimal places in Pi obeys chaos theory, roughly speaking, Pi is chaos in its original form. Can chaos be rational? Certainly! In the same way as the vacuum, with its apparent emptiness, as you know, 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, the American mathematician of Polish origin, Stanislav M. Ulam (it was he who came up with the key idea for the design of a thermonuclear bomb), being present at one very long and very boring (according to him) 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 in a counterclockwise spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Without any ulterior motive, he circled all the prime numbers in black circles along the way. Soon, to his surprise, the circles began to line up along the straight lines with amazing persistence - 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 the brain and the stellar nebula, can be safely 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 corrected by a reasonable number. The above examples show that a reasonable number is also personified on purpose, 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 not, there is not and cannot be a reliable method for determining this, 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 value. Of course, Pi's ideal birth date is March 14, 1592 (3.141592), however, unfortunately, there are no reliable statistics for this year - it is only known that George Villiers Buckingham, the Duke of Buckingham from " Three Musketeers." He was a great swordsman, knew a lot about horses and falconry - but was he Pi? Unlikely. Duncan MacLeod, who was born on March 14, 1592, in the mountains of Scotland, could ideally claim the role of the human embodiment of the number Pi - if he were a real person.
But after all, the year (1592) can be determined according to its own, more logical chronology for Pi. If we accept this assumption, then there are many more applicants 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! And 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 not a lot of coincidences, what do you think?
In what personality Pi is personified today, it is not clear, but in order to see the significance of this number for our world, one does not 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, no doubt, is Pi!
What is a PIN?
Per-SONal IDEN-tifi-KA-ZI-ion number.
What is PI number?
Deciphering the number PI (3, 14 ...) (pin code), anyone can do it without me, through the Glagolitic. We substitute letters instead of numbers (the numerical values of the letters are given in the Glagolitic) and we get the following phrase: Verbs (I say, I say, I do) Az (I, ace, master, creator) Good. And if you take the following numbers, then it turns out something like this: “I do good, I am Fita (hidden, illegitimate child, immaculate conception, unmanifested, 9), I know (know) distortion (evil) this is the speaking (action) will ( desire) The earth I do I know I do the will good evil (distortion) I know evil I do good "..... and so on ad infinitum, there are a lot of numbers, but I believe that everything is about the same thing ...
Music of the number PI
January 13, 2017π= 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 above 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, since 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 πr 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 number is Tau(τ), and if Pi is the ratio of circumference to diameter, then Tau is the ratio of that length to 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."
March 14 is declared the day of the number "Pi", since this date contains the first three digits of this constant.
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 searched for and found in sacred texts.
Who discovered pi?
Who and when first discovered the number π is still a mystery. It is known that the builders of ancient Babylon already used it with might and main when designing. On cuneiform tablets that are thousands of years old, even problems that were proposed to be solved with the help of π have been preserved. True, then it was believed that π is 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 they 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, pi was 3.16.
In ancient India - 3,088.
In Italy, at the turn of the epochs, 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 constructing a square with a compass and straightedge, the area of \u200b\u200bwhich is equal to the area of a certain circle. Archimedes equated π to the fraction 22/7.
The closest to the exact value of π came in China. It was calculated in the 5th century AD. e. famous Chinese astronomer Zu Chun Zhi. Calculating π is quite simple. It was necessary to write odd numbers twice: 11 33 55, and then, dividing them in half, put the first in the denominator of the fraction, and the second in the numerator: 355/113. The result is consistent 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 equals π 3.1415926535 ... and further after the decimal point - to infinity.
The number acquired its designation π in a complicated way: at first, the mathematician Outrade called the circumference with this Greek letter in 1647. He took the first letter of the Greek word περιφέρεια - "periphery". In 1706, the English teacher William Jones, in his Review of the Advances of Mathematics, already called the letter π the ratio of the circumference of a circle to its diameter. And the name was fixed by the 18th-century mathematician Leonhard Euler, before whose authority the rest bowed their heads. So pi became pi.
Number uniqueness
Pi is a truly unique number.
1. Scientists believe that the number of characters 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 Rachmaninov symphony, the Old Testament, your phone number and the year in which the Apocalypse will come.
2. π is related to chaos theory. Scientists came to this conclusion after creating Bailey's computational program, which showed that the sequence of numbers in π is absolutely random, which corresponds to the theory.
3. It is almost impossible to calculate the number to the end - it would take too much time.
4. π is an irrational number, that is, its value cannot be expressed as a fraction.
5. π is a transcendental number. It cannot be obtained by performing any algebraic operations on integers.
6. Thirty-nine decimal places in the number π is enough to calculate the length of a circle encircling known space objects in the Universe, with an error in the radius of a hydrogen atom.
7. The number π is associated with the concept of the "golden section". In the process of measuring the Great Pyramid of Giza, archaeologists found 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 π. It took 23 days, and the mathematician needed a lot of assistants who worked on thousands of computers, united by scattered computing technology. The method allowed making calculations with such a phenomenal speed. It would take more than 500 years to calculate the same on a single computer.
To simply write it all down on paper would require a paper tape over 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 named 67,890 decimal places without making a single mistake.
pi has a lot of fans. It is played on musical instruments, and it turns out that it “sounds” excellently. They remember it and come up with various techniques for this. For the sake of fun, they download it to their computer and brag to each other who downloaded more. 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 interiors. Poems are dedicated to him, he is searched for in holy books and in excavations. There is even a "Club π".
In the best traditions of π, not one, but two whole days a year are devoted to the number! The first time Pi Day is celebrated on March 14th. It is necessary 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.
The second time π 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 π students, schoolchildren and scientists arrange funny flash mobs and actions. Mathematicians, having fun, use π to calculate the laws of a falling sandwich and give each other comic awards.
And by the way, pi can actually be found in holy books. For example, in the Bible. And there the number pi is… three.
There are an infinite number of different numbers in mathematics. Most of them do not attract attention at all. However, some, at first glance, absolutely uninteresting numbers are so well known that they even have their own names. One of these constants is the irrational number Pi, which was studied at school and used to calculate the area or perimeter of a circle along a given radius.
From the history of the constant
Interesting facts about the number Pi - the history of the study. The existence of a constant counts about 4 millennia. In other words, it is a little younger than the science of mathematics itself.
The first evidence that the number pi was known in ancient Egypt is in the papyrus of Ahmes, one of the oldest problem books found. The document dates from approximately 1650 BC. e. In papyrus, the constant was assumed to be 3.1605. This is a fairly accurate value, given that other peoples used 3 to calculate the circumference of a circle from its diameter.
A little more accurately, the number Pi was calculated by Archimedes, the ancient Greek mathematician. He managed to approximate the value in the form of ordinary fractions 22/7 and 223/71. There is a legend that he was so busy calculating the constant that he did not pay attention to how the Romans captured his city. At that moment, when the warrior approached the scientist, Archimedes shouted to him not to touch his drawings. These words of the mathematician were the last.
Al-Khwarizmi, the founder of algebra, who lived in the 8th-9th centuries, worked on the calculations of the constant. With a small error, he received the number Pi, equal to 3.1416.
After 8 centuries, the mathematician Ludolf van Zeulen correctly identified 36 decimal places. For this achievement, the Pi number is sometimes called the Ludolf constant (other well-known names are the Archimedean constant or the circular constant), and the figures obtained by the scientist were engraved on his tombstone.
Around the same time, the constant began to be used not only for a circle, but also for calculating complex curves - arches and hypocycloids.
It was only at the beginning of the 18th century that the constant was called pi. The designation in the form of the letter π was not chosen by chance - it is with it that 2 Greek words begin, meaning circle and perimeter. The name was proposed by the scientist Jones in 1706, and already 30 years later the image of this Greek letter is firmly used among other mathematical notations.
In the 19th century, William Shanks worked on calculating the first 707 characters of a constant. He failed to fully achieve the task - an error crept into the calculations, and the 527 figure turned out to be incorrect. However, even the result obtained was a good achievement for the science of that time.
At the end of the 19th century, the incorrect value of 3.2 was almost accepted at the state level in the state of Indiana. Fortunately, mathematicians managed to oppose the bill and prevent the error.
In the XX-XXI centuries. with the use of computer technology, the accuracy and speed of calculating the constant has increased thousands of times. By 2002, more than 1 trillion digits of the constant had been determined by computer in Japan. After 9 years, the accuracy of the calculation was already 10 trillion characters after the decimal point.
In art and marketing
Even though pi is a mathematical constant, over the years people have tried to use the irrational and mysterious value in other areas of life, including in works of art.
The very first signs of a constant were found in a monument of architecture in Giza. When determining the size of the Great Pyramid, it turned out that the ratio of the perimeter of its base to the height is π. It is only unknown whether the architect wanted to use his knowledge of this number, or whether such a ratio came out by chance.
Currently, the number Pi is also not deprived of attention in creativity. For example, if you mark each note of the minor scale with a number from 0 to 9, and then play the resulting sequence in the form of pi on a musical instrument, you can enjoy an unusual melody with an interesting sound.
Constant also did not bypass the cinema. The drama film Pi: Faith in Chaos won Best Director at the Sundance Film Festival. According to the plot, the main character is in search of simple and understandable answers to questions about the constant, which almost drove him crazy as a result. References to the number are also found in other movies and TV shows.
The number has found its application even in such an unexpected area as marketing. So, the Givenchy company produced a cologne called "Pi".
Constant and Society
Some features of the number:
- The constant is an irrational value. This means that it cannot be represented as a ratio of two numbers. In addition, there is no regularity in his record.
- Characters repeating in a row in a constant are not uncommon. So, for every 20-30 characters, there are usually at least 2 consecutive numbers. Sequences of 3 characters are already rarer, they come across with a frequency of about 1 repetition per 150-300 characters. And on the 763rd sign, a chain of 6 consecutive nines begins. This place in the record even has its own name - the Feynman point.
- If we consider the first million characters, then according to statistics, the rarest numbers in it will be 6 and 1, and the most frequent - 5 and 4.
- The number 0 appears in the sequence later than the rest, only on 31 characters.
- In trigonometry, a 360 degree angle and a constant are closely related. Oddly enough, but at 358, 359 and 360 positions after the decimal point is the number 360.
In order to exchange information about discoveries, the Pi Club was established. Those wishing to join it have to pass a difficult test: a future member of the mathematical community must correctly name as many signs of the constant as possible from memory.
Of course, memorizing a long numerical sequence that does not have patterns and repetitions is a rather difficult task. To facilitate the task, various texts and poems are invented in which the number of letters in a word corresponds to a certain figure of the constant. This method of memorization is popular with members of the Pi Club. One of the longest stories contained 3834 first digits of the number.
Monument at the Museum of Art in Seattle
However, the recognized champions in memorization are, of course, the inhabitants of China and Japan. So, the Japanese Akira Haraguchi was able to learn over 83 thousand digits after the decimal point. And the Chinese Liu Chao became famous as a man who was able to name 67,890 symbols of the number Pi in a record time of 24 hours. At the same time, the average speed was 47 characters per 1 minute. Initially, his goal was to name 93 thousand numbers, but he made a mistake, after which he did not continue.
To emphasize the meaning of the constant, a monument in the form of a huge Greek letter π was erected in front of the Museum of Art in Seattle.
In addition, Pi Day has been celebrated every March 14 since 1988. The date coincides with the first signs of the constant - 3.14. Celebrate it after 1:59. On this day, interested people treat themselves to cakes and cookies with the Pi symbol, after which various mathematical contests and quizzes are held. By the way, it was on this day that A. Einstein, the astronomer Schiaparelli and the astronaut Cernan were born.
The Pi number is an amazing constant that has found its application in a variety of fields, from technology and construction to the arts. Like any other quantity that is used frequently and which cannot be fully calculated, it will always attract the attention of mathematicians, physicists and other scientists.
One of the most mysterious numbers known to mankind, of course, is the number Π (read - pi). In algebra, this number reflects the ratio of the circumference of a circle to its diameter. Previously, this quantity was called the Ludolf number. How and where the number Pi came from is not known for certain, but mathematicians divide the entire history of the number Π into 3 stages, into the ancient, classical and era of digital computers.
The number P is irrational, that is, it cannot be represented as a simple fraction, where the numerator and denominator are integers. Therefore, such a number has no end and is periodic. For the first time, the irrationality of P was proved by I. Lambert in 1761.
In addition to this property, the number P cannot also be the root of any polynomial, and therefore is a number property, when it was proved in 1882, it put an end to the almost sacred dispute of mathematicians “about the squaring of the circle”, which lasted for 2,500 years.
It is known that the first to introduce the designation of this number was the Briton Jones in 1706. After Euler's work appeared, the use of such a designation became generally accepted.
To understand in detail what the number Pi is, it should be said that its use is so widespread that it is difficult to even name a field of science in which it would be dispensed with. One of the simplest and most familiar values from the school curriculum is the designation of the geometric period. The ratio of the length of a circle to the length of its diameter is constant and equal to 3.14. This value was known even to the most ancient mathematicians in India, Greece, Babylon, Egypt. The earliest version of calculating the ratio dates back to 1900 BC. e. A closer to the modern value of P was calculated by the Chinese scientist Liu Hui, in addition, he also invented a quick method for such a calculation. Its value remained generally accepted for almost 900 years.
The classical period in the development of mathematics was marked by the fact that in order to establish exactly what the number Pi is, scientists began to use the methods of mathematical analysis. In the 1400s, the Indian mathematician Madhava used the theory of series to calculate and determined the period of the number P with an accuracy of 11 digits after the decimal point. The first European, after Archimedes, who investigated the number P and made a significant contribution to its justification, was the Dutchman Ludolf van Zeulen, who already determined 15 digits after the decimal point, and wrote very entertaining words in his will: "... whoever is interested - let him go further." It was in honor of this scientist that the number P received its first and only nominal name in history.
The era of computer computing brought new details to the understanding of the essence of the number P. So, in order to find out what the number Pi is, in 1949 the ENIAC computer was used for the first time, one of the developers of which was the future "father" of the theory of modern computers J. The first measurement was carried out on for 70 hours and gave 2037 digits after the decimal point in the period of the number P. The mark of a million characters was reached in 1973. In addition, during this period, other formulas were established that reflect the number P. So, the Chudnovsky brothers were able to find one that made it possible to calculate 1,011,196,691 digits of the period.
In general, it should be noted that in order to answer the question: "What is the number Pi?", Many studies began to resemble competitions. Today, supercomputers are already dealing with the question of what it really is, the number Pi. interesting facts related to these studies permeate almost the entire history of mathematics.
Today, for example, world championships are held in memorizing the number P and world records are set, the latter belongs to the Chinese Liu Chao, who named 67,890 characters in a little over a day. In the world there is even a holiday of the number P, which is celebrated as "Pi Day".
As of 2011, 10 trillion digits of the number period have already been established.
