What is pi? Calculation of the Nth digit of Pi without calculating the previous ones. Secrets of Pi
March 14, 2012
On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: the numerical expression π (Pi) is 3.14 (3rd month (March) 14th).
For the first time, schoolchildren encounter this unusual number in the elementary grades when studying circles and circumferences. The number π is a mathematical constant that expresses the ratio of the circumference of a circle to the length of its diameter. That is, if you take a circle with a diameter equal to one, then the circumference will be equal to the number “Pi”. The number π has an infinite mathematical duration, but in everyday calculations a simplified spelling of the number is used, leaving only two decimal places - 3.14.
In 1987, this day was celebrated for the first time. Physicist Larry Shaw from San Francisco noticed that in American system records of dates (month/day) the date March 14 - 3/14 coincides with the number π (π = 3.1415926...). Typically celebrations begin at 1:59:26 pm (π = 3.14 15926 …).
History of Pi
It is assumed that the history of the number π begins in Ancient Egypt. Egyptian mathematicians determined the area of a circle with diameter D as (D-D/9) 2. From this entry it is clear that at that time the number π was equated to the fraction (16/9) 2, or 256/81, i.e. π 3.160...
In the VI century. BC. in India, in the religious book of Jainism, there are entries indicating that the number π at that time was accepted as equal square root out of 10, which gives the fraction 3.162...
In the 3rd century. BC Archimedes in his short work “Measurement of a Circle” substantiated three propositions:
- Every circle is equal in size right triangle, the legs of which are respectively equal to the length of the circle and its radius;
- The areas of a circle are related to a square built on a diameter as 11 to 14;
- The ratio of any circle to its diameter is less than 3 1/7 and greater than 3 10/71.
Archimedes justified the last position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of circumference to diameter is between the numbers 3 * 10 / 71 and 3 * 1/7, which means that the number “pi” is 3.1419... The true value of this ratio is 3.1415922653...
In the 5th century BC. Chinese mathematician Zu Chongzhi found a more accurate value for this number: 3.1415927...
In the first half of the 15th century. The astronomer and mathematician Kashi calculated π with 16 decimal places.
A century and a half later in Europe, F. Viet found the number π with only 9 regular decimal places: he made 16 doublings of the number of sides of polygons. F. Viet was the first to notice that π can be found using the limits of certain series. This discovery had great importance, it made it possible to calculate π with any accuracy.
In 1706, the English mathematician W. Johnson introduced the notation for the ratio of the circumference of a circle to its diameter and designated it with the modern symbol π, the first letter of the Greek word periferia - circle.
For a long period of time, scientists around the world tried to unravel the mystery of this mysterious number.
What is the difficulty in calculating the value of π?
The number π is irrational: it cannot be expressed as a fraction p/q, where p and q are integers, this number cannot be a root algebraic equation. You cannot specify algebraic or differential equation, the root of which will be π, therefore this number is called transcendental and is calculated by considering any process and is refined by increasing the steps of the process under consideration. Multiple attempts to calculate maximum amount signs of the number π have led to the fact that today, thanks to modern computing technology, it is possible to calculate the sequence with an accuracy of 10 trillion digits after the decimal point.
The digits of the decimal representation of π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that this number contains all written and unwritten books in encrypted form; any information that can be imagined is found in the number π.
You can try to unravel the mystery of this number yourself. Of course, it will not be possible to write down the number “Pi” in full. But for the most curious, I suggest considering the first 1000 digits of the number π = 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
Remember the number "Pi"
Currently, with the help of computer technology, ten trillion digits of the number “Pi” have been calculated. The maximum number of numbers that a person could remember is one hundred thousand.
To remember the maximum number of digits of the number “Pi”, various poetic “memories” are used, in which words with a certain number of letters are arranged in the same sequence as the numbers in the number “Pi”: 3.1415926535897932384626433832795…. To restore the number, you need to count the number of characters in each word and write it down in order.
So I know the number called “Pi”. Well done! (7 digits)
So Misha and Anyuta came running
They wanted to know the number Pi. (11 digits)
This I know and remember perfectly:
And many signs are unnecessary for me, in vain.
Let's trust our enormous knowledge
Those who counted the numbers of the armada. (21 digits)
Once at Kolya and Arina's
We ripped the feather beds.
The white fluff was flying and spinning,
Showered, froze,
Satisfied
He gave it to us
Old women's headache.
Wow, the spirit of fluff is dangerous! (25 characters)
You can use rhyming lines to help you remember the right number.
So that we don't make mistakes,
You need to read it correctly:
Ninety two and six
If you try really hard,
You can immediately read:
Three, fourteen, fifteen,
Ninety two and six.
Three, fourteen, fifteen,
Nine, two, six, five, three, five.
To do science,
Everyone should know this.
You can just try
And repeat more often:
"Three, fourteen, fifteen,
Nine, twenty-six and five."
Still have questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!
On March 14, a very unusual holiday is celebrated all over the world - Pi Day. Everyone has known it since school. Students are immediately explained that the number Pi is a mathematical constant, the ratio of the circumference of a circle to its diameter, which has an infinite value. It turns out that there are many interesting facts associated with this number.
1. The history of numbers goes back more than one thousand years, almost as long as the science of mathematics has existed. Of course, the exact value of the number was not immediately calculated. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, a more accurate measurement was required. By the way, the number existed, but letter designation it only received in early XVII I century (1706) and comes from initial letters two Greek words meaning "circle" and "perimeter". The letter “π” was given to the number by the mathematician Jones, and it became firmly established in mathematics already in 1737.
2. In different eras and among different peoples, the number Pi had different meaning. For example, in Ancient Egypt it was equal to 3.1604, among the Hindus it acquired a value of 3.162, and the Chinese used a number equal to 3.1459. Over time, π was calculated more and more accurately, and when it appeared Computer Engineering, that is, a computer, it began to number more than 4 billion characters.
3. There is a legend, or rather experts believe, that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were wrong. A similar version exists regarding the Temple of Solomon.
4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through law. In 1897, the state of Indiana prepared a bill. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented the mistake. In particular, Professor Perdue, who was present at the legislative meeting, spoke out against the bill.
5. It is interesting that several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after the American physicist. Richard Feynman once gave a lecture and stunned the audience with a remark. He said he wanted to memorize the digits of Pi up to six nines, only to say "nine" six times at the end of the story, implying that its meaning was rational. When in fact it is irrational.
6. Mathematicians around the world do not stop conducting research related to the number Pi. It is literally shrouded in some mystery. Some theorists even believe that it contains universal truth. To exchange knowledge and new information about Pi, a Pi Club was organized. It’s not easy to join; you need to have an extraordinary memory. Thus, those wishing to become a member of the club are examined: a person must recite from memory as many signs of the number Pi as possible.
7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with entire texts. In them, words have the same number of letters as the corresponding number after the decimal point. To make it even easier to remember such a long number, they compose poems according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and intelligence. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) of the first digits of Pi.
8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk managed to recite by heart only two and a half thousand numbers after the decimal point of Pi.
"Pi" in perspective
9. Pi Day has been celebrated for more than a quarter of a century, since 1988. One day, a physicist from the popular science museum in San Francisco, Larry Shaw, noticed that March 14, when written, coincides with the number Pi. In the date, the month and day form 3.14.
10. Pi Day is celebrated not exactly in an original way, but in a fun way. Of course, scientists who occupy positions do not miss it. exact sciences. For them, this is a way not to break away from what they love, but at the same time relax. On this day, people gather and prepare various delicacies with the image of Pi. There is especially room for pastry chefs to roam. They can make cakes with pi written on them and cookies with similar shapes. After tasting the delicacies, mathematicians arrange various quizzes.
11. There is an interesting coincidence. On March 14, the great scientist Albert Einstein, who, as we know, created the theory of relativity, was born. Be that as it may, physicists can also join in the celebration of Pi Day.
Absolutely everyone knows what “pi” is. But the number, familiar to everyone from school, arises 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 other unexpected and far from geometry areas of mathematics. The English mathematician Augustus de Morgan once called pi “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.”
This mysterious number, associated with one of the three classical problems of Antiquity - constructing 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 Ukrainian neurosurgeon, Doctor of Medical Sciences, Professor Andrey Slyusarchuk, who retained 30 million of its characters (20 volumes of text) in memory.
- 3. Did you know that in 1996 Mike Keith wrote short story, which is called “Rhythmic Cadenza” (“Cadeic Cadenze”), in its 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 noticed that on March 14 (in American writing - 3.14) at exactly 01:59, the date and time would coincide with the first digits of the number Pi = 3.14159.
The creator of the theory of relativity, Albert Einstein, was also born on March 14, 1879, which makes this day even more attractive for all mathematics lovers.
In addition, mathematicians also celebrate the day of the approximate value of Pi, which falls on July 22 (22/7 in European date format).
“During this time, they read eulogies in honor of the number Pi and its role in the life of mankind, draw dystopian pictures of a 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 in circles.” , writes Wikipedia.
In numerical terms, Pi begins 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.
Bellar's calculations took him about 103 days. All calculations were carried out on a home computer, the cost of which is around 2000 euros. For comparison, the previous record was set on the T2K Tsukuba System supercomputer, which took about 73 hours to run.
Initially, the number Pi appeared as the ratio of the length 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. Currently, Pi is 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. In total, storing all the numbers requires 1.1 terabytes of disk space.
Such calculations have not only practical significance. 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. At the same time, 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 numbers from 0 to 9 occur in decimal notation Pi an infinite number of times.
In this case, 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.
A number is calculated according to certain rules, and during any calculation, in any place and at any time, the same digit appears at a certain place in the number record. This means that there is a certain law according to which a certain number is placed in a certain place in a number. Of course, this law is not simple, but there is still a law. And this means that the numbers in the number are not random, but logical.
Count the number Pi: PI = 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... - 4/n + 4/(n+2)
Pi search or long division:
Pairs of integers that, when divided, give a close approximation to the number Pi. The division was done in a "column" manner to circumvent the length limitations of Visual Basic 6 floating-point numbers.
Pi = 3.14159265358979323846264>33832795028841 971...
Exotic methods of calculating pi, such as using probability theory 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 do it on a computer) and count the number of ball hits. The software implementation of this model is not yet known
In every book on entertaining mathematics you will certainly find the history of calculating and clarifying 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 Arab mathematicians refined the value of “pi” to 40 digits after the decimal point, and by the beginning of the Computer Age, through the efforts of many enthusiasts, the number of pi had been increased to 500. This accuracy is purely scientific interest(more on this below), for practice, within the Earth, 11 characters after the dot are enough.
Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that if we discard the twelfth digit of “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 when rotating around the Sun (as is known, R = 150 * 106 km = 1.5 * 1014 mm), for the same accuracy it is enough to use “pi” with fourteen digits after the dot. Average distance from the Sun to Pluto, the farthest planet solar system- 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 digits of pi are enough. Why bother about trifles - 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 in the 27th century 34 pi signs were obtained, which are excessive for such distances.
Why is it difficult to calculate the value of pi? The point is that not only is it irrational (that is, it cannot be expressed as a fraction P/Q, where P and Q are integers), but it also cannot 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 (not 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 “simplest” 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 solving the mystery of the number pi, which in purely mathematical terms represents the ratio of the circumference of a circle to its diameter, Der Spiegel reports.
As an irrational quantity, it cannot be represented as a complete fraction, so after the decimal point there is an endless series of digits. This property has always attracted mathematicians who sought to find, on the one hand, a more accurate value of pi, and on the other, its generalized formula.
However, mathematicians David Bailey of the Lawrence Berkeley National Laboratory in California and Richard Grendell 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 decimal numbers. As a result, it was established 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 natural. 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 to unraveling the number pi and, in general, to determining its essence - whether it is normal for our world or not.
Both mathematicians have been interested in pi since 1996, and since that time they have had to abandon the so-called “number theory” and turn their attention to “chaos theory,” which is now their main weapon. Researchers construct, based on the display of 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 truly chaotic, then the series of numbers starting from zero should also be chaotic. But there is no answer to this question yet. The secret of pi, like its older brother - the number 42, with the help of which many researchers are trying to explain the mystery of the universe, has yet to be unraveled."
Interesting data on the distribution of Pi digits.
(Programming is the greatest achievement of mankind. Thanks to it, we regularly learn things that we don’t need to know at all, but are very interesting)
Counted (for a million decimal places):
zeros = 99959,
units = 99758,
twos = 100026,
triples = 100229,
fours = 100230,
fives = 100359,
sixes = 99548,
sevens = 99800,
eight = 99985,
nines = 100106.
In the first 200,000,000,000 decimal places of Pi, the 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, there will be exactly the same number of them, in addition, there will be as many ones as there are twos and threes combined, and even as many as all the other nine digits combined. But here you need to know where to stop, to seize the moment, so to speak, where there are really equal numbers of them.
And one more thing - in the digits of Pi one can expect the appearance of any predetermined sequence of digits. For example, the most common arrangements were found in the following numbers:
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 the number e. (
There was a joke: scientists found the last number in Pi - it turned out to be the number e, they almost got it)
You can search in the first ten thousand digits of Pi for your phone number or date of birth; if that doesn’t work, then look in 100,000 digits.
In the number 1/Pi, starting from 55,172,085,586 digits, there are 33333333333333, isn’t it surprising?
In philosophy, the contingent is usually contrasted with the necessary. So are the signs of pi random? Or are they necessary? Let's say the third digit of pi is "4". And regardless of whoever calculates this pi, in what place and at what time he does it, the third sign will necessarily always be equal to “4”.
The connection between Pi, Phi and the Fibonacci series. The connection between the number 3.1415916 and the number 1.61803 and the Pisa sequence.
- More interesting:
- 1. In the decimal places of Pi, 7, 22, 113, 355 are digit 2. The fractions 22/7 and 355/113 are good approximations to Pi.
- 2. Kokhansky found that Pi is approximate root equations: 9x^4-240x^2+1492=0
- 3. If you write down capital letters 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 of 3,1,4, 1.6 letters.
- (A) BCDEFG (HI) JKL (M) N (O) PQRS (TUVWXY) Z
- 6 3 1 4 1
- So english alphabet must start with the letter H, I or J, not the letter A :)
So what does that matter to us? And it follows from this that in the decimal tail of pi you can find any intended sequence of digits. Your phone number? Please, 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 measly million characters here or take my word for it: any sequence of digits in the decimal places of pi is early or it will be late. Any!
For more elevated readers, we can offer another example: 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 holy books of all religions. I'm not kidding, this is strict 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 that’s it, then that’s it. Including those that correspond to the book you have chosen.
And this again means that it contains not only all world literature, which has already been written (in particular, those books that burned, etc.), but also all the books that still WILL be written.
It turns out that this number (the only reasonable number in the universe!) rules our world.
The question is how to find them there...
And on this day Albert Einstein was born, who predicted... and what didn’t he predict! ...even dark energy.
This world was shrouded in deep darkness.
Let there be light! And then Newton appeared.
But Satan did not wait long for revenge.
Einstein came and everything became the same as before.
They correlate well - pi and albert...
Theories arise, develop and...
The bottom line: Pi is not equal to 3.14159265358979....
This is a misconception based on the erroneous postulate of identifying flat Euclidean space with the real space of the Universe.
A brief explanation of why in general Pi is not equal to 3.14159265358979...
This phenomenon is associated with the curvature of space. The lines of force in the Universe at significant distances are not ideal straight lines, but slightly curved lines. We have already grown to the point of stating the fact that in the real world there are no perfectly straight lines, ideally flat circles, or 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, and certainly not pyramidal. The universe is spherical. The only case when a plane can be ideal (in the sense of “not curved”) is the case 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 we should not neglect the curvature in the orbits of comets and asteroids. The ineradicable Ptolemaic belief that we are still at the center of the Universe can cost us dearly.
Below are the axioms of flat Euclidean (“cubic” Cartesian) space and the additional axiom I formulated for spherical space.
Axioms of flat consciousness:
through 1 point you can draw an infinite number of straight 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.
In the general case, through 3 points it is impossible to draw a single straight line and one, and only one, plane. Additional axiom for spherical consciousness:
In the general case, through 4 points it is impossible to draw a single straight line, a single plane, and one and only one sphere. Arsentiev Alexey Ivanovich
A little mysticism. Is PI Reasonable?
Any other constant can be defined through the number Pi, including the fine structure constant (alpha), the golden proportion constant (f=1.618...), not to mention the number e - this is why the number pi is found not only in geometry, but also in theory of relativity, quantum mechanics, nuclear physics etc. Moreover, scientists have recently found that it is through Pi that one can determine the location elementary particles in the Table of Elementary Particles (previously they tried to do this through Woody’s Table), and the message that in the recently deciphered human DNA, the number Pi is responsible for the DNA structure itself (quite complex, it should be noted), produced the effect of a bomb exploding!
According to Dr. Charles Cantor, under whose leadership DNA was deciphered: “It seems that we have come to the solution to some fundamental problem that the universe has thrown at us. The number Pi is everywhere, it controls all the processes known to us, while remaining unchanged! does the number Pi itself control? There is no answer yet."
In fact, Cantor is disingenuous, there is an answer, it’s just so incredible that scientists prefer not to make it public, fearing for own life(more on this a little later): the number Pi controls itself, it is reasonable! Nonsense? Do not hurry. After all, Fonvizin also said that “in human ignorance, it is very comforting to consider everything that you don’t know as nonsense.”
Firstly, conjectures about the reasonableness of numbers in general have long been visited by many famous mathematicians of our time. Norwegian mathematician Niels Henrik Abel wrote to his mother in February 1829: “I have received confirmation that one of the numbers is reasonable. I spoke with 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, Nils would have revealed the meaning of the number that spoke to him, but on March 6, 1829, he passed away.
1955, Japanese Yutaka Taniyama puts forward the hypothesis that “each elliptic curve corresponds to a certain modular form” (as is known, on the basis of this hypothesis Fermat’s theorem was proven). On September 15, 1955, at an international mathematical symposium in Tokyo, where Taniyama announced his hypothesis, in response to a journalist’s question: “How did you come up with this?” - Taniyama replies: “I didn’t think of it, the number told me about it over the phone.” The journalist, thinking that this was a joke, decided to “support” her: “Did it tell you the phone number?” To which Taniyama seriously replied: “It seems that this number has been known to me for a long time, but I can now report it only after three years, 51 days, 15 hours and 30 minutes.” In November 1958, Taniyama committed suicide. Three years, 51 days, 15 hours and 30 minutes is 3.1415. Coincidence? May be. But here's another one, even stranger. The Italian mathematician Sella Quitino also spent several years, as he vaguely put it, “keeping in touch with one cute number.” The figure, according to Quitino, who was already in a psychiatric hospital at that time, “promised to say his name on his birthday.” Could Quitino have lost his mind so much as to call the number Pi a number, or was he deliberately confusing the doctors? It is not clear, but on March 14, 1827, Quitino passed away.
And the most misterious story associated 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 their research inequalities). As you know, Hardy was officially unmarried, although he repeatedly stated that he was “engaged to the queen of our world.” Fellow scientists more than once heard him talking to someone in his office; no one had ever seen his interlocutor, although his voice - metallic and slightly creaky - had long been the talk of the town at Oxford University, where he worked at last years. In November 1947, these conversations stop, and on December 1, 1947, Hardy is found in a city dump, with a bullet in his stomach. The version of suicide was also confirmed by a note in which Hardy’s hand wrote: “John, you took the queen away from me, I don’t blame you, but I can no longer live without her.”
Is this story related to the number Pi? It’s still unclear, but isn’t it interesting?
Generally speaking, you can collect a lot of similar stories, and, of course, not all of them are tragic.
But, let's move on to "secondly": how can a number even be reasonable? Yes, very simple. The human brain contains 100 billion neurons, the number of decimal places of Pi tends to infinity, in general, according to formal criteria, it can be reasonable. But if you believe the work of the American physicist David Bailey and Canadian mathematicians Peter Borwin and Simon Ploofe, the sequence of decimal places in Pi is subject to chaos theory, roughly speaking, the number Pi is chaos in its original form. Can chaos be intelligent? Certainly! Just like a vacuum, despite its apparent emptiness, as is known, it is by no means empty.
Moreover, if you wish, you can represent this chaos graphically - to make sure that it can be reasonable. In 1965, an American mathematician of Polish origin Stanislaw M. Ulam (he was the one who came up with the key idea for the design of a thermonuclear bomb), while attending one very long and very boring (in his words) meeting, in order to somehow have fun, began to write numbers on checkered paper , included in the number Pi. Putting 3 in the center and moving counterclockwise in a spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Without any second thought, he simultaneously circled all the prime numbers with black circles. Soon, to his surprise, the circles with amazing tenacity began to line up along straight lines - what happened was very similar to something reasonable. Especially after Ulam generated a color picture based on this drawing using a special algorithm.
Actually, this picture, which can be compared with both a brain and a stellar nebula, can safely be called the “brain of Pi.” Approximately with the help of such a structure, this number (the only reasonable number in the universe) controls our world. But how does this control take place? As a rule, with the help of the unwritten laws of physics, chemistry, physiology, astronomy, which are controlled and adjusted by a reasonable number. The above examples show that the intelligent number is also deliberately personified, communicating with scientists as a kind of superpersonality. But if so, did the number Pi come to our world in the guise of an ordinary person?
Complex issue. Maybe it came, maybe it didn’t, there is no reliable method for determining this and there cannot be, but if this number is determined by itself in all cases, then we can assume that it came into our world as a person on the day corresponding to its meaning. Of course, the ideal date of birth for Pi is March 14, 1592 (3.141592), however, unfortunately, there are no reliable statistics for this year - we only know that it was in this year, on March 14, that George Villiers Buckingham, the Duke of Buckingham from " The Three Musketeers." He was an excellent fencer, knew a lot about horses and falconry - but was he Pi? Hardly. Duncan MacLeod, born on March 14, 1592, in the mountains of Scotland, could ideally claim the role of the human embodiment of the number Pi - if he were a real person.
But the year (1592) can be determined according to its own, more logical calendar for Pi. If we accept this assumption, then there are many more candidates for the role of Pi.
The most obvious of them is Albert Einstein, born March 14, 1879. But 1879 is 1592 relative to 287 BC! Why exactly 287? Yes, because it was in this year that Archimedes was born, who for the first time in the world calculated the number Pi as the ratio of the circumference to the diameter and proved that it is the same for any circle! Coincidence? But aren’t there a lot of coincidences, don’t you think?
In what personality Pi is personified today is not clear, but in order to see the meaning of this number for our world, you don’t need to be a mathematician: Pi manifests itself in everything that surrounds us. And this, by the way, is very typical for any intelligent being, which, without a doubt, is Pi!
What is a PIN code?
Per-SONAL IDEN-tifi-KA-CI-on number.
What is PI number?
Decoding the number PI (3, 14...) (pin code), anyone can do this without me, through the Glagolitic alphabet. We substitute letters instead of numbers (the numerical values of the letters are given in Glagolitic) and we get this phrase: Verbs (verb, say, do) Az (I, as, master, creator) Good. And if we take the following numbers, then it turns out something like this: “When I do good, I am Fita (hidden, bastard, immaculate conception, unmanifested, 9), I know (I know) distortion (evil) this is speaking (action) will (desire) Earth I do I know I do will good evil (distortion) I know evil I do good "..... and so on until infinity, there are a lot of numbers, but I believe that everything is about the same thing...
Music of PI
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1. Relevance of the work.
In the infinite variety of numbers, just like among the stars of the Universe, individual numbers and their entire “constellations” of amazing beauty stand out, numbers with extraordinary properties and a unique harmony inherent only to them. You just need to be able to see these numbers and notice their properties. Take a closer look at the natural series of numbers - and you will find in it a lot of surprising and outlandish, funny and serious, unexpected and curious. The one who looks sees. After all, people won’t even notice on a starry summer night... the glow. North Star, if they do not direct their gaze to the cloudless heights.
Moving from class to class, I became acquainted with natural, fractional, decimal, negative, rational. This year I studied irrational. Among irrational numbers There is a special number, the exact calculations of which have been carried out by scientists for many centuries. I came across it back in 6th grade while studying the topic “Circumference and Area of a Circle.” It was emphasized that we would meet with him quite often in classes in high school. Practical tasks on finding the numerical value of π were interesting. The number π is one of most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. The number pi has a lot to do with interesting facts, so it arouses interest in study.
Having heard a lot of interesting things about this number, I myself decided by studying additional literature and searching the Internet to find out as much information as possible about it and answer problematic questions:
How long have people known about the number pi?
Why is it necessary to study it?
What interesting facts are associated with it?
Is it true that the value of pi is approximately 3.14
Therefore, I set myself target: explore the history of the number π and the significance of the number π on modern stage development of mathematics.
Tasks:
Study the literature to obtain information about the history of the number π;
Establish some facts from " modern biography» numbers π;
Practical calculation of the approximate value of the ratio of circumference to diameter.
Object of study:
Object of study: PI number.
Subject of study: Interesting facts related to the PI number.
2. Main part. Amazing number pi.
No other number is as mysterious as Pi, with its famous never-ending number series. In many areas of mathematics and physics, scientists use this number and its laws.
Of all the numbers used in mathematics, science, engineering, and Everyday life, is given as much attention as is given to the number pi. One book says, “Pi is captivating the minds of science geniuses and amateur mathematicians around the world” (“Fractals for the Classroom”).
It can be found in probability theory, in solving problems with complex numbers and other unexpected and far from geometry areas of mathematics. The English mathematician Augustus de Morgan once called pi “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.” This mysterious number, associated with one of the three classical problems of Antiquity - constructing a square whose area is equal to the area of a given circle - entails a trail of dramatic historical and curious entertaining facts.
Some even consider it one of the five most important numbers in mathematics. But as the book Fractals for the Classroom notes, as important as pi is, “it is difficult to find areas in scientific calculations that require more than twenty decimal places of pi.”
3. The concept of pi
The number π is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. The number π (pronounced "pi") is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. Denoted by the letter "pi" of the Greek alphabet.
In numerical terms, π begins as 3.141592 and has an infinite mathematical duration.
4. History of the number "pi"
According to experts, this number was discovered by Babylonian magicians. It was used in the construction of the famous Tower of Babel. However, the insufficiently accurate calculation of the value of Pi led to the collapse of the entire project. It is possible that this mathematical constant underlay the construction of the legendary Temple of King Solomon.
The history of pi, which expresses the ratio of the circumference of a circle to its diameter, began in Ancient Egypt. Area of a circle with diameter d Egyptian mathematicians defined it as (d-d/9) 2 (this entry is given here in modern symbols). From the above expression we can conclude that at that time the number p was considered equal to the fraction (16/9) 2 , or 256/81 , i.e. π = 3,160...
In the holy book of Jainism (one of ancient religions, which existed in India and arose in the 6th century. BC) there is an indication from which it follows that the number p at that time was taken equal, which gives the fraction 3,162... Ancient Greeks Eudoxus, Hippocrates and others reduced the measurement of a circle to the construction of a segment, and the measurement of a circle to the construction of an equal square. It should be noted that for many centuries mathematicians different countries and peoples tried to express the ratio of the circumference to the diameter as a rational number.
Archimedes in the 3rd century BC. in his short work “Measuring a Circle” he substantiated three propositions:
Every circle is equal in size to a right triangle, the legs of which are respectively equal to the length of the circle and its radius;
The areas of a circle are related to the square built on the diameter, as 11 to 14;
The ratio of any circle to its diameter is less 3 1/7 and more 3 10/71 .
According to exact calculations Archimedes the ratio of circumference to diameter is enclosed between the numbers 3*10/71 And 3*1/7 , which means that π = 3,1419... The true meaning of this relationship 3,1415922653... In the 5th century BC. Chinese mathematician Zu Chongzhi a more accurate value for this number was found: 3,1415927...
In the first half of the 15th century. observatory Ulugbek, near Samarkand, astronomer and mathematician al-Kashi calculated pi to 16 decimal places. Al-Kashi made unique calculations that were needed to compile a table of sines in steps of 1" . These tables played important role in astronomy.
A century and a half later in Europe F. Viet found pi with only 9 correct decimal places by doubling the number of sides of polygons 16 times. But at the same time F. Viet was the first to notice that pi can be found using the limits of certain series. This discovery was of great
value, since it allowed us to calculate pi with any accuracy. Only 250 years after al-Kashi his result was surpassed.
Birthday of the number “”.
The unofficial holiday “PI Day” is celebrated on March 14, which in American format (day/date) is written as 3/14, which corresponds to the approximate value of PI.
There is an alternative version of the holiday - July 22. It's called Approximate Pi Day. The fact is that representing this date as a fraction (22/7) also gives the number Pi as a result. It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who noticed that the date and time coincided with the first digits of the number π.
Interesting facts related to the number “”
Scientists at the University of Tokyo, led by Professor Yasumasa Kanada, managed to set a world record in calculating the number Pi to 12,411 trillion digits. To do this, a group of programmers and mathematicians needed a special program, a supercomputer and 400 hours of computer time. (Guinness Book of Records).
The German king Frederick II was so fascinated by this number that he dedicated to it... the entire palace of Castel del Monte, in the proportions of which PI can be calculated. Now the magical palace is under the protection of UNESCO.
How to remember the first digits of the number “”.
The first three digits of the number = 3.14... are not difficult to remember. And for remembering more signs there are funny sayings and poems. For example, these:
You just have to try
And remember everything as it is:
Ninety two and six.
S. Bobrov. "Magic bicorn"
Anyone who learns this quatrain will always be able to name 8 signs of the number :
In the following phrases, the number signs can be determined by the number of letters in each word:
“What do I know about circles?” (3.1416);
“So I know the number called Pi. - Well done!"
(3,1415927);
“Learn and know the number behind the number, how to notice good luck.”
(3,14159265359)
5. Notation for pi
The first to introduce the modern symbol pi for the ratio of the circumference of a circle to its diameter was an English mathematician W.Johnson in 1706. As a symbol he took the first letter of the Greek word "periphery", which translated means "circle". Entered W.Johnson the designation became commonly used after the publication of the works L. Euler, who used the entered character for the first time in 1736 G.
At the end of the 18th century. A.M.Lagendre based on works I.G. Lambert proved that pi is irrational. Then the German mathematician F. Lindeman based on research S.Ermita, found strict proof that this number is not only irrational, but also transcendental, i.e. cannot be the root of an algebraic equation. The search for the exact expression for pi continued after the work F. Vieta. At the beginning of the 17th century. Dutch mathematician from Cologne Ludolf van Zeijlen(1540-1610) (some historians call him L.van Keulen) found 32 correct signs. Since then (year of publication 1615), the value of the number p with 32 decimal places has been called the number Ludolph.
6. How to remember the number "Pi" accurate to eleven digits
The number "Pi" is the ratio of the circumference of a circle to its diameter, it is expressed as infinite decimal. In everyday life, it is enough for us to know three signs (3.14). However, some calculations require greater accuracy.
Our ancestors did not have computers, calculators or reference books, but since the time of Peter I they have been engaged in geometric calculations in astronomy, mechanical engineering, and shipbuilding. Subsequently, electrical engineering was added here - there is the concept of “circular frequency” alternating current". To remember the number "Pi" a couplet was invented (unfortunately, we do not know the author and the place of its first publication; but back in the late 40s of the twentieth century, Moscow schoolchildren studied from Kiselev's geometry textbook, where it was given).
The couplet is written according to the rules of old Russian orthography, according to which after consonant must be placed at the end of the word "soft" or "solid" sign. Here it is, this wonderful historical couplet:
Who, jokingly, will soon wish
“Pi” knows the number - he already knows.
It makes sense for anyone who plans to engage in precise calculations in the future to remember this. So what is the number "Pi" accurate to eleven digits? Count the number of letters in each word and write these numbers in a row (separate the first number with a comma).
This accuracy is already quite sufficient for engineering calculations. In addition to the ancient one, there is also modern way memorization, which was pointed out by a reader who identified himself as Georgy:
So that we don't make mistakes,
You need to read it correctly:
Three, fourteen, fifteen,
Ninety two and six.
You just have to try
And remember everything as it is:
Three, fourteen, fifteen,
Ninety two and six.
Three, fourteen, fifteen,
Nine, two, six, five, three, five.
To do science,
Everyone should know this.
You can just try
And repeat more often:
"Three, fourteen, fifteen,
Nine, twenty-six and five."
Well, mathematicians with the help of modern computers can calculate almost any number of digits of Pi.
7. Pi memory record
Humanity has been trying to remember the signs of pi for a long time. But how to put infinity into memory? A favorite question of professional mnemonists. Many unique theories and techniques for mastering a huge amount of information have been developed. Many of them have been tested on pi.
The world record set in the last century in Germany is 40,000 characters. The Russian record for pi values was set on December 1, 2003 in Chelyabinsk by Alexander Belyaev. In an hour and a half with short breaks, Alexander wrote 2500 digits of pi on the blackboard.
Before this, listing 2,000 characters was considered a record in Russia, which was achieved in 1999 in Yekaterinburg. According to Alexander Belyaev, head of the center for the development of figurative memory, any of us can conduct such an experiment with our memory. It is only important to know special memorization techniques and practice periodically.
Conclusion.
The number pi appears in formulas used in many fields. Physics, electrical engineering, electronics, probability theory, construction and navigation are just a few. And it seems that just as there is no end to the signs of the number pi, there is no end to the possibilities for the practical application of this useful, elusive number pi.
In modern mathematics, the number pi is not only the ratio of the circumference to the diameter, it is included in big number various formulas.
This and other interdependencies allowed mathematicians to further understand the nature of pi.
The exact value of the number π in modern world represents not only its own scientific value, but is also used for very accurate calculations(for example, satellite orbits, construction of giant bridges), as well as estimates of the speed and power of modern computers.
Currently, the number π is associated with a difficult-to-see set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this speaks of a growing interest in the most important mathematical constant, the study of which dates back more than twenty-two centuries.
The work I did was interesting. I wanted to know about the history of the number pi, practical application and I think I achieved my goal. Summarizing the work, I come to the conclusion that this topic relevant. There are many interesting facts associated with the number π, so it arouses interest in study. In my work, I became more familiar with number - one of the eternal values that humanity has been using for many centuries. I learned some aspects of its rich history. Found out why ancient world did not know the correct ratio of circumference to diameter. I looked clearly at the ways in which the number can be obtained. Based on experiments, I calculated the approximate value of the number different ways. Processed and analyzed the experimental results.
Any schoolchild today should know what a number means and approximately equals. After all, everyone’s first acquaintance with a number, its use in calculating the circumference of a circle, the area of a circle, occurs in the 6th grade. But, unfortunately, this knowledge remains formal for many and after a year or two, few people remember not only that the ratio of the length of a circle to its diameter is the same for all circles, but they even have difficulty remembering the numerical value of the number, equal to 3 ,14.
I tried to lift the veil of the rich history of the number that humanity has been using for many centuries. I made a presentation for my work myself.
The history of numbers is fascinating and mysterious. I would like to continue researching other amazing numbers in mathematics. This will be the subject of my next research studies.
Bibliography.
1. Glazer G.I. History of mathematics in school, grades IV-VI. - M.: Education, 1982.
2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook - M.: Prosveshchenie, 1989.
3. Zhukov A.V. The ubiquitous number “pi”. - M.: Editorial URSS, 2004.
4. Kympan F. History of the number “pi”. - M.: Nauka, 1971.
5. Svechnikov A.A. a journey into the history of mathematics - M.: Pedagogika - Press, 1995.
6. Encyclopedia for children. T.11.Mathematics - M.: Avanta +, 1998.
Internet resources:
- http:// crow.academy.ru/materials_/pi/history.htm
Http://hab/kp.ru// daily/24123/344634/