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, the area of \u200b\u200bwhich 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, like 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, having discarded the twelfth digit "pi" after the point when calculating the length of the meridian, we 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, there’s nothing to trifle about - 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 a seemingly chaotic series of digits 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 became 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 point 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 you 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. The Norwegian mathematician Niels 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 frightens 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, the 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 a journalist's question: "How did you think of that?" - 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, this is how 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? Hardly. 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 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
PI
The symbol PI stands for the ratio of the circumference of a circle to its diameter. For the first time in this sense, the symbol p was used by W. Jones in 1707, and L. Euler, having accepted this designation, introduced it into scientific use. Even in ancient times, mathematicians knew that calculating the value of p and the area of a circle are closely related tasks. The ancient Chinese and ancient Jews considered the number p equal to 3. The value of p, equal to 3.1605, is contained in the ancient Egyptian papyrus of the scribe Ahmes (c. 1650 BC). Around 225 BC e. Archimedes, using regular 96-gons inscribed and circumscribed, approximated the area of a circle using a method that resulted in a PI value between 31/7 and 310/71. Another approximate value of p, equivalent to the usual decimal representation of this number 3.1416, has been known since the 2nd century. L. van Zeulen (1540-1610) calculated the value of PI with 32 decimal places. By the end of the 17th century. new methods of mathematical analysis made it possible to calculate the value of p in many different ways. In 1593 F. Viet (1540-1603) derived the formula
In 1665 J. Wallis (1616-1703) proved that
In 1658, W. Brounker found a representation of the number p in the form of a continued fraction
G. Leibniz in 1673 published a series
Series allow you to calculate the value of p with any number of decimal places. In recent years, with the advent of electronic computers, the value of p has been found with more than 10,000 digits. With ten digits, the value of PI is 3.1415926536. As a number, PI has some interesting properties. For example, it cannot be represented as a ratio of two integers or as a periodic decimal; the number PI is transcendental, i.e. cannot be represented as a root of an algebraic equation with rational coefficients. The PI number is included in many mathematical, physical and technical formulas, including those not directly related to the area of a circle or the length of an arc of a circle. For example, the area of an ellipse A is given by A = pab, where a and b are the lengths of the major and minor semiaxes.
Collier Encyclopedia. - Open Society. 2000 .
See what "PI NUMBER" is in other dictionaries:
number- Reception Source: GOST 111 90: Sheet glass. Specifications original document See also related terms: 109. Number of betatron oscillations ... Dictionary-reference book of terms of normative and technical documentation
Ex., s., use. very often Morphology: (no) what? numbers for what? number, (see) what? number than? number about what? about the number; pl. What? numbers, (no) what? numbers for what? numbers, (see) what? numbers than? numbers about what? about mathematics numbers 1. Number ... ... Dictionary of Dmitriev
NUMBER, numbers, pl. numbers, numbers, numbers, cf. 1. A concept that serves as an expression of quantity, something with the help of which objects and phenomena are counted (mat.). Integer. A fractional number. named number. Prime number. (see simple1 in 1 value).… … Explanatory Dictionary of Ushakov
An abstract designation, devoid of special content, of any member of a certain series, in which this member is preceded or followed by some other definite member; an abstract individual feature that distinguishes one set from ... ... Philosophical Encyclopedia
Number- Number is a grammatical category that expresses the quantitative characteristics of objects of thought. The grammatical number is one of the manifestations of a more general linguistic category of quantity (see the Linguistic category) along with a lexical manifestation (“lexical ... ... Linguistic Encyclopedic Dictionary
A number approximately equal to 2.718, which is often found in mathematics and science. For example, during the decay of a radioactive substance after time t, a fraction equal to e kt remains from the initial amount of substance, where k is a number, ... ... Collier Encyclopedia
A; pl. numbers, villages, slam; cf. 1. A unit of account expressing one or another quantity. Fractional, integer, simple hours. Even, odd hours. Count as round numbers (approximately, counting as whole units or tens). Natural hours (positive integer ... encyclopedic Dictionary
Wed quantity, count, to the question: how much? and the very sign expressing quantity, the figure. Without number; no number, no count, many many. Put the appliances according to the number of guests. Roman, Arabic or church numbers. Integer, contra. fraction. ... ... Dahl's Explanatory Dictionary
NUMBER, a, pl. numbers, villages, slam, cf. 1. The basic concept of mathematics is the value, with the help of which the swarm is calculated. Integer hours Fractional hours Real hours Complex hours Natural hours (positive integer). Simple hours (natural number, not ... ... Explanatory dictionary of Ozhegov
NUMBER "E" (EXP), an irrational number that serves as the basis of natural LOGARITHMS. This real decimal number, an infinite fraction equal to 2.7182818284590...., is the limit of the expression (1/) as n goes to infinity. In fact,… … Scientific and technical encyclopedic dictionary
Quantity, cash, composition, strength, contingent, amount, figure; day.. Wed. . See day, quantity. a small number, no number, grow in number... Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M .: Russians ... ... Synonym dictionary
Books
- Name number. Secrets of numerology. Exit from the body for the lazy. ESP Primer (number of volumes: 3), Lawrence Shirley. Name number. Secrets of numerology. Shirley B. Lawrence's book is a comprehensive study of the ancient esoteric system - numerology. To learn how to use number vibrations to…
- Name number. The sacred meaning of numbers. Symbolism of the Tarot (number of volumes: 3), Uspensky Petr. Name number. Secrets of numerology. Shirley B. Lawrence's book is a comprehensive study of the ancient esoteric system - numerology. To learn how to use number vibrations to…
What is Pi hiding?
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 a circle, that is, the impossibility of constructing a square with a compass and straightedge, the area of which 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 club
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.
MUNICIPAL BUDGET EDUCATIONAL INSTITUTION "NOVOAGANSKAYA COMPREHENSIVE SECONDARY SCHOOL №2"
History of occurrence
pi numbers.
Performed by Shevchenko Nadezhda,
student 6 "B" class
Head: Chekina Olga Alexandrovna, teacher of mathematics
town Novoagansk
2014
Plan.
- Doing.
Goals.
II. Main part.
1) The first step to the number pi.
2) An unsolved mystery.
3) Interesting facts.
III. Conclusion
References.
Introduction
Goals of my work
1) Find the history of the origin of pi.
2) Tell interesting facts about pi
3) Make a presentation and issue a report.
4) Prepare a speech for the conference.
Main part.
Pi (π) is the letter of the Greek alphabet used in mathematics to denote the ratio of the circumference of a circle to its diameter. This designation comes from the initial letter of the Greek words περιφέρεια - circle, periphery and περίμετρος - perimeter. It became generally accepted after the work of L. Euler, referring to 1736, but for the first time it was used by the English mathematician W. Jones (1706). Like any irrational number, π is represented by an infinite non-periodic decimal fraction:
π = 3.141592653589793238462643.
The first step in studying the properties of the number π was made by Archimedes. In the essay "Measurement of the circle" he derived the famous inequality: [formula]
This means that π lies in an interval of length 1/497. In the decimal number system, three correct significant digits are obtained: π \u003d 3.14 .... Knowing the perimeter of a regular hexagon and successively doubling the number of its sides, Archimedes calculated the perimeter of a regular 96-gon, from which follows the inequality. A 96-gon visually differs little from a circle and is a good approximation to it.
In the same work, successively doubling the number of sides of a square, Archimedes found the formula for the area of a circle S = π R2. Later, he also supplemented it with the formulas for the area of a sphere S = 4 π R2 and the volume of a ball V = 4/3 π R3.
In ancient Chinese writings come across a variety of estimates, of which the most accurate is the well-known Chinese number 355/113. Zu Chongzhi (5th century) even considered this value to be accurate.
Ludolf van Zeulen (1536-1610) spent ten years calculating the number π with 20 decimal digits (this result was published in 1596). Applying the method of Archimedes, he brought doubling to an n-gon, where n=60 229. Having outlined his results in the essay “On the Circumference”, Ludolf ended it with the words: “Whoever has a desire, let him go further.” After his death, 15 more exact digits of the number π were discovered in his manuscripts. Ludolph bequeathed that the signs he found were carved on his tombstone. In honor of him, the number π was sometimes called the "Ludolf number".
But the mystery of the mysterious number has not been resolved until today, although it still worries scientists. Attempts by mathematicians to completely calculate the entire numerical sequence often lead to curious situations. For example, the mathematicians the Chudnovsky brothers at the Polytechnic University of Brooklyn designed a super-fast computer specifically for this purpose. However, they failed to set a record - while the record belongs to the Japanese mathematician Yasumasa Kanada, who was able to calculate 1.2 billion numbers in an infinite sequence.
Interesting Facts
The unofficial holiday "Pi Day" is celebrated on March 14, which in American date format (month / day) is written as 3/14, which corresponds to the approximate value of Pi.
Another date associated with the number π is July 22, which is called the “Approximate Pi Day”, since in the European date format this day is written as 22/7, and the value of this fraction is an approximate value of the number π.
The world record for memorizing the signs of the number π belongs to the Japanese Akira Haraguchi (Akira Haraguchi). He memorized the number pi up to the 100,000th decimal place. It took him almost 16 hours to name the whole number.
The German king Frederick the Second was so fascinated by this number that he dedicated to it ... the whole palace of Castel del Monte, in the proportions of which Pi can be calculated. Now the magical palace is under the protection of UNESCO.
Conclusion
At present, the number π is associated with an incomprehensible set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this indicates a growing interest in the most important mathematical constant, the study of which has been going on for more than twenty-two centuries.
My work can be used in mathematics lessons.
Results of my work:
- Found the history of the origin of the number pi.
- She talked about interesting facts about the number pi.
- Learned a lot about pi.
- Designed the work and spoke at the conference.
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 A 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 signs.
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 discovered by Euler between the exponential and trigonometric functions, 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
