On March 14, a very unusual holiday is celebrated all over the world - Pi Day. Everyone has known it since school days. 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 a lot of interesting facts are connected with this number.
1. The history of number has more than one millennium, almost as long as the science of mathematics exists. 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 it received a letter designation only in early XVIII century (1706) and comes from the initial letters of two Greek words meaning "circumference" and "perimeter". The mathematician Jones endowed the number with the letter "π", and she firmly entered 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 the value of 3.162, the Chinese used the 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 have more than 4 billion characters.
3. There is a legend, more precisely, 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 mistaken. A similar version exists regarding Solomon's temple.
4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through the law. In 1897, a bill was drafted in the state of Indiana. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented an error. In particular, Professor Purdue, who was present at the legislative assembly, 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 an American physicist. Once Richard Feynman was giving a lecture and stunned the audience with a remark. He said he wanted to learn the digits of pi up to six nines by heart, only to say "nine" six times at the end of the story, hinting that its meaning was rational. When in fact it is irrational.
6. Mathematicians around the world do not stop doing research related to the number Pi. It is literally shrouded in mystery. Some theorists even believe that it contains a universal truth. In order to share knowledge and new information about Pi, they organized the Pi Club. Entering it is not easy, you need to have an outstanding memory. So, those wishing to become a member of the club are examined: a person must tell as many signs of the number Pi from memory 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 whole texts. In them, words have the same number of letters as the corresponding digit after the decimal point. To further simplify the memorization of such a long number, they compose verses 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 ingenuity. 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) 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 was able to memorize 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 was spelled the same as pi. In a date, the month and day form 3.14.
10. Pi Day is celebrated not only in an original way, but in a fun way. Of course, scientists who occupy exact sciences. For them, this is a way not to break away from what they love, but at the same time to relax. On this day, people gather and cook different goodies with the image of Pi. Especially there is a place for confectioners to roam. They can make pi cakes and similarly shaped cookies. After tasting the treats, mathematicians arrange various quizzes.
11. There is an interesting coincidence. On March 14, the great scientist Albert Einstein was born, who, as you know, created the theory of relativity. Be that as it may, physicists can also join in the celebration of Pi Day.
There are a lot of mysteries among the PIs. Rather, these are not even riddles, but a kind of some kind of Truth that no one has yet figured out in the entire history of mankind ...
What is Pi? The PI number is a mathematical "constant" that expresses the ratio of the circumference of a circle to its diameter. At first, due to ignorance, it (this ratio) was considered equal to three, which was roughly approximate, but they were enough. But when prehistoric times were replaced by ancient times (that is, already historical), then there was no limit to the surprise of inquisitive minds: it turned out that the number three very inaccurately expresses this ratio. With the passage of time and the development of science, this number began to be considered equal to twenty-two-sevenths.
The English mathematician August de Morgan once called the number PI "... mysterious number 3.14159… that crawls through the door, through the window, and through the roof.” Tireless scientists continued and continued to calculate the decimal places of the number Pi, which is actually a wildly non-trivial task, because you can’t just calculate it in a column: the number is not only irrational, but also transcendental (these are just such numbers that are not calculated by simple equations).
In the process of calculating these very signs, many different scientific methods and entire sciences were discovered. But the most important thing is that there are no repetitions in the decimal part of the number pi, as in an ordinary periodic fraction, and the number of decimal places in it is infinite. To date, it has been verified that there really are no repetitions in 500 billion digits of the number pi. There are reasons to believe that they do not exist at all.
Since there are no repetitions in the sequence of signs of the number pi, this means that the sequence of signs of the number pi obeys chaos theory, more precisely, the number pi is chaos written in numbers. Moreover, if desired, this chaos can be represented graphically, and there is an assumption that this Chaos is reasonable.
In 1965, the American mathematician M. Ulam, sitting at a boring meeting, from nothing to do, began to write numbers included in the number pi on checkered paper. 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. Along the way, he circled all the prime numbers. What was his surprise and horror when the circles began to line up along the straight lines!
In the decimal tail of pi, you can find any conceived sequence of digits. Any sequence of digits in decimal places of pi will sooner or later be found. Any!
So what? - you ask. And then. Estimate: if your phone is there (and it is), then there is also the phone of the girl who did not want to give you her number. Moreover, there are also credit card numbers, and even all the values of the winning numbers of tomorrow's lottery draw. Why, in general, all lotteries for many millennia to come. The question is how to find them there ...
If you encrypt all the letters in 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. This is a hard scientific fact. After all, the sequence is INFINITE and combinations in the number PI 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 world literature, which has already been written (in particular, those books that burned down, etc.), but also all the books that WILL be written. Including your articles on the sites. It turns out that this number (the only reasonable number in the Universe!) controls our world. You just need to consider more signs, find the right area and decipher it. This is something akin to a paradox with a herd of chimpanzees hammering on the keyboard. With a long enough (one can even estimate this time) experiment, they will print all of Shakespeare's plays.
This immediately suggests an analogy with periodically appearing reports that the Old Testament allegedly encoded messages to posterity that can be read with the help of ingenious programs. It is not entirely wise to dismiss such an exotic feature of the Bible right off the bat, caballists have been searching for such prophecies for centuries, but I would like to cite the message of one researcher who, using a computer, found in the Old Testament the words that there are no prophecies in the Old Testament. Most likely, in a very large text, as well as in the infinite digits of the number PI, you can not only encode any information, but also “find” phrases that were not originally included there.
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 we, having discarded the twelfth digit in the number of PI after the point when calculating the length of the meridian, will be mistaken by several millimeters. And when calculating the length of the Earth's orbit during rotation around the Sun (as you know, R \u003d 150 * 106 km \u003d 1.5 * 1014 mm), for the same accuracy, it is enough to use the number PI with fourteen digits after the point, but what’s there to trifle - the diameter of our Galaxies are about 100,000 light years (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm. and they are currently calculated to 12411 trillion signs!!!
The absence of periodically repeating figures, namely, based on their formula Circumference = Pi * D, the circle does not close, since there is no finite number. This fact can also be closely related to the spiral manifestation in our lives...
There is also a hypothesis that all (or some) universal constants (Planck's constant, Euler's number, universal gravitational constant, electron charge, etc.) change their values over time, as the curvature of space changes due to the redistribution of matter or for other reasons unknown to us.
At the risk of incurring the wrath of the enlightened community, we can assume that the PI number considered today, which reflects the properties of the Universe, can change over time. In any case, no one can forbid us to re-find the value of the number PI, confirming (or not confirming) the existing values.
10 interesting facts about the number of pi
1. The history of number has more than one millennium, almost as long as the science of mathematics exists. 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 it received a letter designation only at the beginning of the 18th century (1706) and comes from the initial letters of two Greek words meaning “circumference” and “perimeter”. The mathematician Jones endowed the number with the letter "π", and she firmly entered mathematics already in 1737.
2. In different eras and among different peoples, the number Pi had different meanings. For example, in ancient Egypt it was 3.1604, among the Hindus it acquired the value of 3.162, the Chinese used the number equal to 3.1459. Over time, π was calculated more and more accurately, and when computer technology appeared, that is, a computer, it began to have more than 4 billion characters.
3. There is a legend, more precisely, 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 mistaken. A similar version exists regarding Solomon's temple.
4. It is noteworthy that they tried to introduce the value of the number Pi even at the state level, that is, through the law. In 1897, a bill was drafted in the state of Indiana. Pi was 3.2 according to the document. However, scientists intervened in time and thus prevented an error. In particular, Professor Purdue, who was present at the legislative assembly, spoke out against the bill.
5. Interestingly, several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after an American physicist. Once Richard Feynman was giving a lecture and stunned the audience with a remark. He said he wanted to learn the digits of pi up to six nines by heart, only to say "nine" six times at the end of the story, hinting that its meaning was rational. When in fact it is irrational.
6. Mathematicians around the world do not stop doing research related to the number Pi. It is literally shrouded in mystery. Some theorists even believe that it contains a universal truth. In order to share knowledge and new information about Pi, they organized the Pi Club. Entering it is not easy, you need to have an outstanding memory. So, those wishing to become a member of the club are examined: a person must tell as many signs of the number Pi from memory 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 whole texts. In them, words have the same number of letters as the corresponding digit after the decimal point. To further simplify the memorization of such a long number, they compose verses 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 ingenuity. 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) 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 was able to memorize only two and a half thousand numbers after the decimal point of Pi.
9. Pi Day has been celebrated for more than a quarter of a century, since 1988. Once, a physicist from the Popular Science Museum in San Francisco, Larry Shaw, noticed that March 14 was spelled the same as pi. In a date, the month and day form 3.14.
10. There is an interesting coincidence. On March 14, the great scientist Albert Einstein was born, who, as you know, created the theory of relativity.
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 factorial, in solving problems with complex numbers and 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 is mysterious number, associated with one of the three classical tasks of Antiquity - the construction of a square, the area of \u200b\u200bwhich is equal to the area of a given circle - entails a train of dramatic historical and curious fun 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 short story, which is called "Rhythmic Cadenze" ("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 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 eulogies 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, decide math puzzles and riddles, as well as 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.
At first Pi was calculated in binary system, after which it was 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 each book entertaining mathematics you will certainly find the history of the calculation and refinement of 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 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 "pi" signs are enough. Yes, what is there to trifle - the diameter of our Galaxy is about 100,000 light years (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm., And back in the 27th century, 34 pi signs were obtained, redundant for such distances.
What is the complexity of calculating the value of "pi"? The fact is that it is not only irrational (that is, it cannot be expressed as a fraction P / Q, where P and Q are integers), but it cannot yet be a root algebraic equation. A number, for example, an irrational one, cannot be represented by a ratio of integers, but it is the root of the equation X2-2=0, and for the numbers "pi" and e (Euler's constant), such an algebraic (non-differential) equation cannot be specified. Such numbers (transcendental) are calculated by considering a process and are refined by increasing the steps of the process under consideration. The most “simple” way is to inscribe a regular polygon in a circle and calculate the ratio of the perimeter of the polygon to its “radius”...pages marsu
Number explains the world
It seems that two American mathematicians have managed to get closer to unraveling the mystery of the number pi, which in purely mathematical terms represents the ratio of the circumference of a circle to its diameter, reports Der Spiegel.
As an irrational value, it cannot be represented as a complete fraction, so an endless series of numbers follows the decimal point. This property has always attracted mathematicians who sought to find, on the one hand, a more accurate value of pi, and, on the other hand, its generalized formula.
However, mathematicians David Bailey of the Lawrence Berkeley National Laboratory in California and Richard Grendel of Reed College in Portland looked at the number from a different angle - they tried to find some meaning in the seemingly chaotic series of numbers after the decimal point. As a result, it was found that combinations of the following numbers are regularly repeated - 59345 and 78952.
But so far they cannot answer the question of whether the repetition is random or regular. The question of the pattern of repetition of certain combinations of numbers, and not only in the number pi, is one of the most difficult in mathematics. But now we can say something more definite about this number. The discovery paves the way for unraveling the number pi and, in general, for determining its essence - whether it is normal for our world or not.
Both mathematicians have been interested in the number pi since 1996, and since that time they have had to abandon the so-called "number theory" and pay attention to the "chaos theory", which is now their main weapon. Researchers construct based on the display of the number pi - its most common form is 3.14159 ... - series of numbers between zero and one - 0.314, 0.141, 0.415, 0.159 and so on. Therefore, if the number pi is indeed chaotic, then the series of numbers starting from zero must also be chaotic. But there is no answer to this question yet. To unravel the secret of pi, like its older brother - the number 42, with the help of which many researchers are trying to explain the secret of the universe, has yet to be."
Interesting data about the distribution of pi digits.
(Programming is the greatest achievement of mankind. Thanks to it, we regularly learn what we don’t need to know at all, but it’s very interesting)
Calculated (for a million decimal places):
zeros = 99959,
units = 99758,
twos = 100026,
triplets = 100229,
fours = 100230,
fives = 100359,
sixes = 99548,
sevens = 99800,
eights = 99985,
nines = 100106.
In the first 200,000,000,000 decimal places of pi, digits occurred with the following frequency:
"0" : 20000030841;
"1" : 19999914711;
"2" : 20000136978;
"3" : 20000069393
"4" : 19999921691;
"5" : 19999917053;
"6" : 19999881515;
"7" : 19999967594
"8" : 20000291044;
"9" : 19999869180;
That is, the numbers are distributed almost evenly. Why? Because according to modern mathematical concepts, with an infinite number of digits, they will be exactly equal, in addition, there will be as many ones as twos and triples combined, and even as many as all the other nine digits combined. But here to know where to stop, to seize the moment, so to speak, where they are really evenly divided.
And yet - in the digits of Pi, you can expect the appearance of any predetermined sequence of digits. For example, the most common arrangements were found in the following numbers in a row:
01234567891: from 26.852.899.245
01234567891: from 41,952,536,161
01234567891: from 99.972.955.571
01234567891: from 102,081,851,717
01234567891: from 171,257,652,369
01234567890: from 53,217,681,704
27182818284: c 45,111,908,393 are the digits of e. (
There was such a joke: scientists found last number in the record 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 capital letters of the English alphabet clockwise into a circle and cross out the letters having 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 lit.
- (A) BCDEFG (HI) JKL (M) N (O) PQRS (TUVWXY) Z
- 6 3 1 4 1
- So that 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 general case Pi is not equal to 3.14159265358979...
This phenomenon is associated with the curvature of space. lines of force in the Universe at considerable distances are not ideal straight lines, but slightly curved lines. We have already matured to the point of stating the fact that in real world there are no perfectly straight lines, no perfectly flat circles, no perfect 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 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 number Pi is responsible for the very structure of DNA (quite complicated, it should be noted), produced the effect of a bombshell!
According to Dr. Charles Cantor, under whose leadership DNA was deciphered: “It seems that we have come to the solution of some fundamental problem that the universe has thrown at us. The number Pi is everywhere, it controls all the processes known to us, while remaining unchanged! does it control Pi itself? There is no answer yet."
In fact, Kantor is cunning, there is an answer, it’s just so incredible that scientists prefer not to make it public, fearing for their own lives (more on that later): Pi controls itself, it is reasonable! Nonsense? Do not hurry. After all, even Fonvizin said that "in human ignorance it is very comforting to consider everything as nonsense that you do not know."
First, conjectures about the reasonableness of numbers in general have long visited many famous mathematicians of our time. Norwegian mathematician Nils Henrik Abel wrote to his mother in February 1829: “I received confirmation that one of the numbers is reasonable. I talked to him! But it scares me that I cannot determine what this number is. But maybe this is for the best. The Number warned me that I would be punished if It was revealed." Who knows, Niels would have revealed the meaning of the number that spoke to him, but on March 6, 1829, he died.
1955, Japanese Yutaka Taniyama puts forward the hypothesis that "every elliptic curve corresponds to a certain modular form" (as is known, Fermat's theorem was proved on the basis of this hypothesis). September 15, 1955, at the International Mathematical Symposium in Tokyo, where Taniyama announced his conjecture, to 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, 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 last 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 (he owns key idea design of a thermonuclear bomb), being present at one very long and very boring (according to him) meeting, in order to somehow have fun, he began to write the numbers included in the number Pi on checkered paper. 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 Pi - if he was 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. Substitute letters instead of numbers ( numerical values letters are given in the Glagolitic) and we get this phrase: Verbs (I speak, I say, I do) Az (I, ace, master, creator) Good. And if you take the following numbers, then it turns out something like this: “I do good, I am Fita (hidden, illegitimate child, immaculate conception, unmanifested, 9), I know (know) distortion (evil) this is the speaking (action) will ( desire) The earth I do I know I do the will good evil (distortion) I know evil I do good "..... and so on ad infinitum, there are a lot of numbers, but I believe that everything is about the same thing ...
Music of the number PI
What is the number pi we know and remember from school. It is equal to 3.1415926 and so on... To an ordinary person it is enough to know that this number is obtained by dividing the circumference of a circle by its diameter. But many people know that the number Pi appears in unexpected areas not only in mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you can see a lot of surprises among the endless series of numbers. Is it possible that Pi hides the deepest secrets of the universe?
Infinite number
The number Pi itself arises in our world as the circumference of a circle, the diameter of which equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi starts like 3.1415926 and goes to infinity in rows of numbers that never repeat. First amazing fact is that this number, used in geometry, cannot be expressed as a fraction of whole numbers. In other words, you cannot write it as a ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no such equation (polynomial) with integer coefficients, the solution of which would be Pi.
The fact that the number Pi is transcendent was proved in 1882 by the German mathematician von Lindemann. It was this proof that became the answer to the question whether it is possible to draw a square with a compass and a ruler, whose area is equal to the area of a given circle. This problem is known as the search for the squaring of a circle, which has troubled mankind since ancient times. It seemed that this problem had a simple solution and was about to be revealed. But it was an incomprehensible property of pi that showed that the problem of squaring a circle has no solution.
For at least four and a half millennia, mankind has been trying to get an increasingly accurate value of pi. For example, in the Bible in the 1st Book of Kings (7:23), the number pi is taken equal to 3.
Remarkable in accuracy, the value of Pi can be found in the pyramids of Giza: the ratio of the perimeter and height of the pyramids is 22/7. This fraction gives an approximate value of Pi, equal to 3.142 ... Unless, of course, the Egyptians set such a ratio by accident. The same value already in relation to the calculation of the number Pi was received in the III century BC by the great Archimedes.
In the Ahmes Papyrus, an ancient Egyptian mathematics textbook that dates back to 1650 BC, Pi is calculated as 3.160493827.
In ancient Indian texts around the 9th century BC, the most accurate value was expressed by the number 339/108, which equaled 3.1388 ...
For almost two thousand years after Archimedes, people have been trying to find ways to calculate pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Mark Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Ariabhata, the medieval mathematician Leonardo of Pisa, known as Fibonacci, the Arab scientist Al-Khwarizmi, from whose name the word "algorithm" appeared. All of them and many other people were looking for the most accurate methods for calculating Pi, but until the 15th century they never received more than 10 digits after the decimal point due to the complexity of the calculations.
Finally, in 1400, the Indian mathematician Madhava from the Sangamagram calculated Pi with an accuracy of up to 13 digits (although he still made a mistake in the last two).
Number of signs
In the 17th century, Leibniz and Newton discovered the analysis of infinitesimal quantities, which made it possible to calculate pi more progressively - through power series and integrals. Newton himself calculated 16 decimal places, but did not mention this in his books - this became known after his death. Newton claimed that he only calculated Pi out of boredom.
At about the same time, other lesser-known mathematicians also pulled themselves up, proposing new formulas for calculating the number Pi through trigonometric functions.
For example, here is the formula used to calculate Pi by astronomy teacher John Machin in 1706: PI / 4 = 4arctg(1/5) - arctg(1/239). Using methods of analysis, Machin derived from this formula the number Pi with a hundred decimal places.
By the way, in the same 1706, the number Pi received an official designation in the form of a Greek letter: it was used by William Jones in his work on mathematics, taking the first letter of the Greek word “periphery”, which means “circle”. Born in 1707, the great Leonhard Euler popularized this designation, which is now known to any schoolchild.
Before the era of computers, mathematicians were concerned with calculating as many signs as possible. In this regard, sometimes there were curiosities. Amateur mathematician W. Shanks calculated 707 digits of pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoveries in Paris in 1937. However, nine years later, observant mathematicians found that only the first 527 characters were correctly calculated. The museum had to incur decent expenses to correct the mistake - now all the numbers are correct.
When computers appeared, the number of digits of Pi began to be calculated in completely unimaginable orders.
One of the first electronic computers ENIAC, created in 1946, which was huge and generated so much heat that the room warmed up to 50 degrees Celsius, calculated the first 2037 digits of Pi. This calculation took the car 70 hours.
As computers improved, our knowledge of pi went further and further into infinity. In 1958, 10 thousand digits of the number were calculated. In 1987, the Japanese calculated 10,013,395 characters. In 2011, Japanese researcher Shigeru Hondo passed the 10 trillion mark.
Where else can you find Pi?
So, often our knowledge of the number Pi remains at the school level, and we know for sure that this number is indispensable in the first place in geometry.
In addition to the formulas for the length and area of a circle, the number Pi is used in the formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: somewhere the formulas are simple and easy to remember, and somewhere they contain very complex integrals.
Then we can meet the number pi in mathematical formulas, where, at first glance, geometry is not visible. For example, the indefinite integral of 1/(1-x^2) is Pi.
Pi is often used in series analysis. For example, here is a simple series that converges to pi:
1/1 - 1/3 + 1/5 - 1/7 + 1/9 - .... = PI/4
Among series, pi appears most unexpectedly in the well-known Riemann zeta function. It will not be possible to tell about it in a nutshell, we will only say that someday the number Pi will help to find a formula for calculating prime numbers.
And it is absolutely amazing: Pi appears in two of the most beautiful "royal" formulas of mathematics - the Stirling formula (which helps to find the approximate value of the factorial and the gamma function) and the Euler formula (which relates as many as five mathematical constants).
However, the most unexpected discovery awaited mathematicians in probability theory. Pi is also there.
For example, the probability that two numbers are relatively prime is 6/PI^2.
Pi appears in Buffon's 18th-century needle-throwing problem: what is the probability that a needle thrown onto a sheet of paper with a pattern will cross one of the lines. If the length of the needle is L, and the distance between the lines is L, and r > L, then we can approximately calculate the value of Pi using the probability formula 2L/rPI. Just imagine - we can get Pi from random events. And by the way Pi is present in the normal probability distribution, appears in the equation of the famous Gaussian curve. Does this mean that pi is even more fundamental than just the ratio of a circle's circumference to its diameter?
We can meet Pi in physics as well. Pi appears in Coulomb's law, which describes the force of interaction between two charges, in Kepler's third law, which shows the period of revolution of a planet around the Sun, and even occurs in the arrangement of electron orbitals of a hydrogen atom. And, again, the most incredible thing is that the Pi number is hidden in the formula of the Heisenberg uncertainty principle, the fundamental law of quantum physics.
Secrets of Pi
In Carl Sagan's novel "Contact", which is based on the film of the same name, aliens inform the heroine that among the signs of Pi there is a secret message from God. From a certain position, the numbers in the number cease to be random and represent a code in which all the secrets of the Universe are recorded.
This novel actually reflected the riddle that occupies the minds of mathematicians all over the planet: is the number Pi a normal number in which the digits are scattered with the same frequency, or is there something wrong with this number. And although scientists tend to the first option (but cannot prove it), Pi looks very mysterious. A Japanese man once calculated how many times the numbers from 0 to 9 occur in the first trillion digits of pi. And I saw that the numbers 2, 4 and 8 are more common than the rest. This may be one of the hints that Pi is not quite normal, and the numbers in it are really not random.
Let's remember everything that we have read above and ask ourselves, what other irrational and transcendental number is so common in the real world?
And there are other oddities in store. For example, the sum of the first twenty digits of Pi is 20, and the sum of the first 144 digits is equal to the "number of the beast" 666.
The protagonist of the American TV series The Suspect, Professor Finch, told students that, due to the infinity of pi, any combination of numbers can occur in it, from the numbers of your date of birth to more complex numbers. For example, in the 762nd position there is a sequence of six nines. This position is called the Feynman point, after the famous physicist who noticed this interesting combination.
We also know that the number Pi contains the sequence 0123456789, but it is located on the 17,387,594,880th digit.
All this means that in the infinity of the number Pi you can find not only interesting combinations of numbers, but also the encoded text of "War and Peace", the Bible, and even the main secret Universe, if it exists.
By the way, about the Bible. The well-known popularizer of mathematics Martin Gardner in 1966 stated that the millionth sign of the number Pi (at that time still unknown) would be the number 5. He explained his calculations by the fact that in the English version of the Bible, in the 3rd book, 14th chapter, 16 -m verse (3-14-16) the seventh word contains five letters. The million figure was received eight years later. It was number five.
Is it worth it after this to assert that the number pi is random?
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 by the set various 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 fraction; 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 Dmitrieva
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. 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 grammatical category expressing quantitative characteristics 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 lexical manifestation("lexical ... ... Linguistic Encyclopedic Dictionary
A number approximately equal to 2.718, which is often found in mathematics and natural sciences. 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
BUT; 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 h. ( natural number, not… … Explanatory dictionary of Ozhegov
NUMBER "E" (EXP), an irrational number that serves as the basis natural logarithms. It's valid decimal number, an infinite fraction equal to 2.7182818284590...., is the limit of the expression (1/) as n tends 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…
