There is no cyclicity and system in the digits after the decimal point, that is, in the decimal expansion of Pi there is any sequence of digits that you can imagine (including a very rare sequence of a million non-trivial zeros in mathematics, predicted by the German mathematician Bernhardt Riemann back in 1859).
This means that Pi, in coded form, contains all written and unwritten books, and in general any information that exists (which is why the calculations of the Japanese professor Yasumasa Kanada, who recently determined the number Pi to 12411 trillion decimal places, were right there classified - with such a volume of data it is not difficult to recreate the contents of any secret document printed before 1956, although this data is not enough to determine the location of any person, this requires at least 236734 trillion decimal places - it is assumed that such work is now being carried out in Pentagon (using quantum computers, the clock frequency of processors of which is already approaching the sound speed today).
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 is found not only in geometry, but also in the theory of relativity , quantum mechanics, nuclear physics, etc. Moreover, scientists have recently found that it is through Pi that one can determine the location of elementary particles in the Table of elementary particles (previously they tried to do this through the Woody Table), and the message that in the recently deciphered human DNA, the Pi number is responsible for the DNA structure itself (enough complex, it should be noted), produced the effect of an exploding bomb!
According to Dr. Charles Cantor, under whose leadership DNA was deciphered: “It seems that we have come to unraveling some fundamental puzzle that the universe has thrown at us. The number Pi is everywhere, it controls all the processes known to us, while remaining unchanged! Who controls the Pi itself? No response 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 don’t know.
First, conjectures about the reasonableness of numbers in general have long visited many famous mathematicians of our time. The Norwegian mathematician Niels Henrik Abel wrote to his mother in February 1829: “I have received confirmation that one of the numbers is reasonable. I spoke to him! But it scares me that I can't figure out what that number is. But maybe that's for the best. The Number warned me that I would be punished if It was revealed.” Who knows, Niels would have revealed the meaning of the number that spoke to him, but on March 6, 1829, he died.
1955, the Japanese Yutaka Taniyama puts forward the hypothesis that “every elliptic curve corresponds to a certain modular form” (as is known, Fermat's theorem was proved on the basis of this hypothesis). September 15, 1955, at the International Mathematical Symposium in Tokyo, where Taniyama announced his conjecture, to the question of a journalist: “How did you think of this?” - Taniyama replies: “I didn’t 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 give you a 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 number.” The figure, according to Kvitino, who was already in a psychiatric hospital at that time, “promised to tell her name on her birthday.” Could Kvitino have lost his mind so much as to call the number Pi a number, or was he deliberately confusing doctors? It is not clear, but on March 14, 1827, Kvitino died.
And the most mysterious story is connected with the “great Hardy” (as you all know, this is how contemporaries called the great English mathematician Godfrey Harold Hardy), who, together with his friend John Littlewood, is famous for his work in number theory (especially in the field of Diophantine approximations) and function theory ( where friends became famous for the study of inequalities). As you know, Hardy was officially unmarried, although he repeatedly stated that he was "betrothed to the queen of our world." Fellow scientists have heard him talking to someone in his office more than once, no one has ever seen his interlocutor, although his voice - metallic and slightly raspy - has long been the talk of the town at Oxford University, where he worked in recent years . In November 1947, these conversations stop, and on December 1, 1947, Hardy is found in the city dump, with a bullet in his stomach. The version of suicide was also confirmed by a note, where Hardy's hand was written: "John, you stole the queen from me, I don't blame you, but I can no longer live without her."
Is this story related to pi? So far it is unclear, but isn't it curious?+
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
Borwin and Simon Plofe, the sequence of decimal places in Pi is subject to chaos theory, roughly speaking, Pi is chaos in its original form. Can chaos be rational? Certainly! In the same way as the vacuum, with its apparent emptiness, as you know, it is by no means empty.
Moreover, if you wish, you can represent this chaos graphically - to make sure that it can be reasonable. In 1965, the American mathematician of Polish origin Stanislav M. Ulam (it was he who came up with the key idea for the design of a thermonuclear bomb), being present at one very long and very boring (according to him) meeting, in order to somehow have fun, began to write numbers on checkered paper , included in the number Pi.
Putting 3 in the center and moving in a counterclockwise spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Without any ulterior motive, he circled all the prime numbers in black circles along the way. Soon, to his surprise, the circles began to line up along the straight lines with amazing persistence - what happened was very similar to something reasonable. Especially after Ulam generated a color picture based on this drawing, using a special algorithm.
Actually, this picture, which can be compared with both the brain and the stellar nebula, can be safely called the “brain of Pi”. Approximately with the help of such a structure, this number (the only reasonable number in the universe) controls our world. But how does this control take place? As a rule, with the help of the unwritten laws of physics, chemistry, physiology, astronomy, which are controlled and corrected by a reasonable number. The above examples show that a reasonable number is also personified on purpose, communicating with scientists as a kind of superpersonality. But if so, did the number Pi come to our world, in the guise of an ordinary person?
Difficult question. Maybe it came, maybe not, there is not and cannot be a reliable method for determining this, but if this number is determined by itself in all cases, then we can assume that it came into our world as a person on the day corresponding to its value. Of course, Pi's ideal birth date is March 14, 1592 (3.141592), however, unfortunately, there are no reliable statistics for this year - it is only known that George Villiers Buckingham, the Duke of Buckingham from “ Three Musketeers." He was a great swordsman, knew a lot about horses and falconry - but was he Pi? Hardly. Duncan MacLeod, who was born on March 14, 1592, in the mountains of Scotland, could ideally claim the role of the human embodiment of the number Pi - if he were a real person.
But after all, the year (1592) can be determined according to its own, more logical chronology for Pi. If we accept this assumption, then there are many more applicants for the role of Pi.
The most obvious of them is Albert Einstein, born March 14, 1879. But 1879 is 1592 relative to 287 BC! And why exactly 287? Yes, because it was in this year that Archimedes was born, who for the first time in the world calculated the number Pi as the ratio of the circumference to the diameter and proved that it is the same for any circle!
Coincidence? But not a lot of coincidences, what do you think?
In what personality Pi is personified today, it is not clear, but in order to see the significance of this number for our world, one does not need to be a mathematician: Pi manifests itself in everything that surrounds us. And this, by the way, is very typical for any intelligent being, which, no doubt, is Pi!
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 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 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 involved in the exact sciences do not miss it. 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.
Introduction
The article contains mathematical formulas, so for reading go to the site for their correct display. The number \(\pi \) has a rich history. This constant denotes the ratio of the circumference of a circle to its diameter.
In science, the number \(\pi \) is used in any calculation where there are circles. Starting from the volume of a can of soda, to the orbits of satellites. And not just circles. Indeed, in the study of curved lines, the number \(\pi \) helps to understand periodic and oscillatory systems. For example, electromagnetic waves and even music.
In 1706, in the book "A New Introduction to Mathematics" by the British scientist William Jones (1675-1749), the letter of the Greek alphabet \(\pi\) was used for the first time to denote the number 3.141592.... This designation comes from the initial letter of the Greek words περιϕερεια - circle, periphery and περιµετρoς - perimeter. The generally accepted designation became after the work of Leonhard Euler in 1737.
geometric period
The constancy of the ratio of the length of any circle to its diameter has been noticed for a long time. The inhabitants of Mesopotamia used a rather rough approximation of the number \(\pi \). As follows from ancient problems, they use the value \(\pi ≈ 3 \) in their calculations.
A more precise value for \(\pi \) was used by the ancient Egyptians. In London and New York, two parts of an ancient Egyptian papyrus are kept, which is called the "Rhinda Papyrus". The papyrus was compiled by the scribe Armes between about 2000-1700 BC. BC. Armes wrote in his papyrus that the area of a circle with a radius \(r\) is equal to the area of a square with a side equal to \(\frac(8)(9) \) from the diameter of the circle \(\frac(8 )(9) \cdot 2r \), i.e. \(\frac(256)(81) \cdot r^2 = \pi r^2 \). Hence \(\pi = 3,16\).
The ancient Greek mathematician Archimedes (287-212 BC) first set the task of measuring a circle on a scientific basis. He got the score \(3\frac(10)(71)< \pi < 3\frac{1}{7}\), рассмотрев отношение периметров вписанного и описанного 96-угольника к диаметру окружности. Архимед выразил приближение числа \(\pi \) в виде дроби \(\frac{22}{7}\), которое до сих называется архимедовым числом.
The method is quite simple, but in the absence of ready-made tables of trigonometric functions, root extraction will be required. In addition, the approximation to \(\pi \) converges very slowly: with each iteration, the error only decreases by a factor of four.
Analytical period
Despite this, until the middle of the 17th century, all attempts by European scientists to calculate the number \ (\ pi \) were reduced to increasing the sides of the polygon. For example, the Dutch mathematician Ludolf van Zeilen (1540-1610) calculated the approximate value of the number \(\pi \) with an accuracy of 20 decimal digits.
It took him 10 years to figure it out. By doubling the number of sides of inscribed and circumscribed polygons according to the method of Archimedes, he came up with \(60 \cdot 2^(29) \) - a gon in order to calculate \(\pi \) with 20 decimal places.
After his death, 15 more exact digits of the number \(\pi \) were found in his manuscripts. Ludolph bequeathed that the signs he found were carved on his tombstone. In honor of him, the number \(\pi \) was sometimes called the "Ludolf number" or the "Ludolf constant".
One of the first to introduce a method different from that of Archimedes was François Viet (1540-1603). He came to the result that a circle whose diameter is equal to one has an area:
\[\frac(1)(2 \sqrt(\frac(1)(2)) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1 )(2)) ) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt (\frac(1)(2) \cdots )))) \]
On the other hand, the area is \(\frac(\pi)(4) \). Substituting and simplifying the expression, we can obtain the following infinite product formula for calculating the approximate value \(\frac(\pi)(2) \):
\[\frac(\pi)(2) = \frac(2)(\sqrt(2)) \cdot \frac(2)(\sqrt(2 + \sqrt(2))) \cdot \frac(2 )(\sqrt(2+ \sqrt(2 + \sqrt(2)))) \cdots \]
The resulting formula is the first exact analytical expression for the number \(\pi \). In addition to this formula, Viet, using the method of Archimedes, gave with the help of inscribed and circumscribed polygons, starting with a 6-gon and ending with a polygon with \(2^(16) \cdot 6 \) sides, an approximation of the number \(\pi \) with 9 correct signs.
The English mathematician William Brounker (1620-1684) used the continued fraction to calculate \(\frac(\pi)(4)\) as follows:
\[\frac(4)(\pi) = 1 + \frac(1^2)(2 + \frac(3^2)(2 + \frac(5^2)(2 + \frac(7^2 )(2 + \frac(9^2)(2 + \frac(11^2)(2 + \cdots )))))) \]
This method of calculating the approximation of the number \(\frac(4)(\pi) \) requires quite a lot of calculations to get at least a small approximation.
The values obtained as a result of the substitution are either greater or less than the number \(\pi \), and each time closer to the true value, but getting the value 3.141592 will require quite a large calculation.
Another English mathematician John Machin (1686-1751) in 1706 used the formula derived by Leibniz in 1673 to calculate the number \(\pi \) with 100 decimal places, and applied it as follows:
\[\frac(\pi)(4) = 4 arctg\frac(1)(5) - arctg\frac(1)(239) \]
The series converges quickly and can be used to calculate the number \(\pi \) with great accuracy. Formulas of this type were used to set several records in the computer age.
In the 17th century with the beginning of the period of mathematics of variable magnitude, a new stage began in the calculation of \(\pi \). The German mathematician Gottfried Wilhelm Leibniz (1646-1716) in 1673 found the expansion of the number \(\pi \), in general form it can be written as the following infinite series:
\[ \pi = 1 - 4(\frac(1)(3) + \frac(1)(5) - \frac(1)(7) + \frac(1)(9) - \frac(1) (11) + \cdots) \]
The series is obtained by substituting x = 1 into \(arctg x = x - \frac(x^3)(3) + \frac(x^5)(5) - \frac(x^7)(7) + \frac (x^9)(9) - \cdots\)
Leonhard Euler develops the idea of Leibniz in his work on the use of series for arctg x when calculating the number \(\pi \). The treatise "De variis modis circuli quadraturam numeris proxime exprimendi" (On the various methods of expressing the squaring of a circle by approximate numbers), written in 1738, discusses methods for improving calculations using the Leibniz formula.
Euler writes that the arc tangent series will converge faster if the argument tends to zero. For \(x = 1\) the convergence of the series is very slow: to calculate with an accuracy of up to 100 digits, it is necessary to add \(10^(50)\) terms of the series. You can speed up calculations by decreasing the value of the argument. If we take \(x = \frac(\sqrt(3))(3)\), then we get the series
\[ \frac(\pi)(6) = artctg\frac(\sqrt(3))(3) = \frac(\sqrt(3))(3)(1 - \frac(1)(3 \cdot 3) + \frac(1)(5 \cdot 3^2) - \frac(1)(7 \cdot 3^3) + \cdots) \]
According to Euler, if we take 210 terms of this series, we get 100 correct digits of the number. The resulting series is inconvenient, because it is necessary to know a sufficiently precise value of the irrational number \(\sqrt(3)\). Also, in his calculations, Euler used expansions of arc tangents into the sum of arc tangents of smaller arguments:
\[where x = n + \frac(n^2-1)(m-n), y = m + p, z = m + \frac(m^2+1)(p) \]
Far from all the formulas for calculating \(\pi \) that Euler used in his notebooks have been published. In published works and notebooks, he considered 3 different series for calculating the arc tangent, and also made many statements regarding the number of summable terms needed to obtain an approximate value \(\pi \) with a given accuracy.
In subsequent years, the refinement of the value of the number \(\pi \) happened faster and faster. So, for example, in 1794, George Vega (1754-1802) already identified 140 signs, of which only 136 turned out to be correct.
Computing period
The 20th century was marked by a completely new stage in the calculation of the number \(\pi \). The Indian mathematician Srinivasa Ramanujan (1887-1920) discovered many new formulas for \(\pi\). In 1910, he obtained a formula for calculating \(\pi \) through the expansion of the arc tangent in a Taylor series:
\[\pi = \frac(9801)(2\sqrt(2) \sum\limits_(k=1)^(\infty) \frac((1103+26390k) \cdot (4k){(4\cdot99)^{4k} (k!)^2}} .\]!}
With k=100, an accuracy of 600 correct digits of the number \(\pi \) is achieved.
The advent of computers made it possible to significantly increase the accuracy of the obtained values in a shorter period of time. In 1949, using ENIAC, a group of scientists led by John von Neumann (1903-1957) obtained 2037 decimal places of \(\pi \) in just 70 hours. David and Gregory Chudnovsky in 1987 obtained a formula with which they were able to set several records in the calculation \(\pi \):
\[\frac(1)(\pi) = \frac(1)(426880\sqrt(10005)) \sum\limits_(k=1)^(\infty) \frac((6k)!(13591409+545140134k ))((3k)!(k!)^3(-640320)^(3k)).\]
Each member of the series gives 14 digits. In 1989, 1,011,196,691 decimal places were received. This formula is well suited for calculating \(\pi \) on personal computers. At the moment, the brothers are professors at the Polytechnic Institute of New York University.
An important recent development was the discovery of the formula in 1997 by Simon Pluff. It allows you to extract any hexadecimal digit of the number \(\pi \) without calculating the previous ones. The formula is called the "Bailey-Borwain-Pluff formula" in honor of the authors of the article where the formula was first published. It looks like this:
\[\pi = \sum\limits_(k=1)^(\infty) \frac(1)(16^k) (\frac(4)(8k+1) - \frac(2)(8k+4 ) - \frac(1)(8k+5) - \frac(1)(8k+6)) .\]
In 2006, Simon, using PSLQ, came up with some nice formulas for computing \(\pi \). For example,
\[ \frac(\pi)(24) = \sum\limits_(n=1)^(\infty) \frac(1)(n) (\frac(3)(q^n - 1) - \frac (4)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]
\[ \frac(\pi^3)(180) = \sum\limits_(n=1)^(\infty) \frac(1)(n^3) (\frac(4)(q^(2n) - 1) - \frac(5)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]
where \(q = e^(\pi)\). In 2009, Japanese scientists, using the T2K Tsukuba System supercomputer, obtained the number \(\pi \) with 2,576,980,377,524 decimal places. The calculations took 73 hours 36 minutes. The computer was equipped with 640 four-core AMD Opteron processors, which provided a performance of 95 trillion operations per second.
The next achievement in calculating \(\pi \) belongs to the French programmer Fabrice Bellard, who at the end of 2009 on his personal computer running Fedora 10 set a record by calculating 2,699,999,990,000 decimal places of the number \(\pi \). Over the past 14 years, this is the first world record set without the use of a supercomputer. For high performance, Fabrice used the formula of the Chudnovsky brothers. In total, the calculation took 131 days (103 days of calculation and 13 days of verification). Bellar's achievement showed that for such calculations it is not necessary to have a supercomputer.
Just six months later, François' record was broken by engineers Alexander Yi and Singer Kondo. To set a record of 5 trillion decimal places \(\pi \), a personal computer was also used, but with more impressive characteristics: two Intel Xeon X5680 processors at 3.33 GHz, 96 GB of RAM, 38 TB of disk memory and operating system Windows Server 2008 R2 Enterprise x64. For calculations, Alexander and Singer used the formula of the Chudnovsky brothers. The calculation process took 90 days and 22 TB of disk space. In 2011, they set another record by calculating 10 trillion decimal places for the number \(\pi \). The calculations took place on the same computer that had set their previous record and took a total of 371 days. At the end of 2013, Alexander and Singeru improved the record to 12.1 trillion digits of the number \(\pi \), which took them only 94 days to calculate. This improvement in performance is achieved by optimizing software performance, increasing the number of processor cores, and significantly improving software fault tolerance.
The current record is that of Alexander Yi and Singeru Kondo, which is 12.1 trillion decimal places of \(\pi \).
Thus, we examined the methods for calculating the number \(\pi \) used in ancient times, analytical methods, and also examined modern methods and records for calculating the number \(\pi \) on computers.
List of sources
- Zhukov A.V. The ubiquitous number Pi - M.: LKI Publishing House, 2007 - 216 p.
- F. Rudio. On the squaring of the circle, with an appendix of the history of the question, compiled by F. Rudio. / Rudio F. - M .: ONTI NKTP USSR, 1936. - 235c.
- Arndt, J. Pi Unleashed / J. Arndt, C. Haenel. - Springer, 2001. - 270p.
- Shukhman, E.V. Approximate calculation of Pi using a series for arctg x in published and unpublished works by Leonhard Euler / E.V. Shukhman. - History of science and technology, 2008 - No. 4. - P. 2-17.
- Euler, L. De variis modis circuliaturam numeris proxime exprimendi/ Commentarii academiae scientiarum Petropolitanae. 1744 - Vol. 9 - 222-236p.
- Shumikhin, S. Number Pi. History of 4000 years / S. Shumikhin, A. Shumikhina. — M.: Eksmo, 2011. — 192p.
- Borwein, J.M. Ramanujan and Pi. / Borwein, J.M., Borwein P.B. In the world of science. 1988 - No. 4. - S. 58-66.
- Alex Yee. number world. Access mode: numberworld.org
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The meaning of the number "Pi", as well as its symbolism, is known all over the world. This term denotes irrational numbers (that is, their value cannot be accurately expressed as a fraction y / x, where y and x are integers) and is borrowed from the ancient Greek phraseological unit "peripheria", which can be translated into Russian as "circle".The number "Pi" in mathematics denotes the ratio of the circumference of a circle to the length of its diameter. The history of the origin of the number "Pi" goes into the distant past. Many historians have tried to establish when and by whom this symbol was invented, but they failed to find out.
Pi" is a transcendental number, or, in simple terms, it cannot be the root of some polynomial with integer coefficients. It can be denoted as a real number or as an indirect number that is not algebraic.
Pi is 3.1415926535 8979323846 2643383279 5028841971 6939937510...
Pi" can be not only an irrational number that cannot be expressed using several different numbers. The number "Pi" can be represented by a certain decimal fraction, which has an infinite number of digits after the decimal point. Another interesting point - all these numbers are not able to repeat.
Pi" can be correlated with the fractional number 22/7, the so-called "triple octave" symbol. This number was known even by ancient Greek priests. In addition, even ordinary residents could use it to solve any everyday problems, as well as use it to design such complex structures as tombs.
According to the scientist and researcher Hayens, a similar number can be traced among the ruins of Stonehenge, and also found in the Mexican pyramids.
Pi" mentioned in his writings Ahmes, a well-known engineer at that time. He tried to calculate it as accurately as possible by measuring the diameter of a circle from the squares drawn inside it. Probably, in a certain sense, this number has a certain mystical, sacred meaning for the ancients.
Pi" in fact, is the most mysterious mathematical symbol. It can be classified as a delta, omega, etc. It is such a relationship that will be exactly the same, regardless of which point in the universe the observer will be. In addition, it will be unchanged from the measurement object.
Most likely, the first person who decided to calculate the number "Pi" using the mathematical method is Archimedes. He decided he was drawing regular polygons in a circle. Considering the diameter of the circle as a unit, the scientist denoted the perimeter of the polygon drawn in the circle, considering the perimeter of the inscribed polygon as an upper estimate, but as a lower estimate of the circumference
What is the number "Pi"
The text of the work is placed without images and formulas.
The full version of the work is available in the "Job Files" tab in PDF format
1. The relevance of the work.
In an infinite number of numbers, as well as among the stars of the Universe, separate numbers and their whole “constellations” of amazing beauty stand out, numbers with unusual properties and a peculiar harmony inherent only to them. You just need to be able to see these numbers, notice their properties. Look closely at the natural series of numbers - and you will find in it a lot of amazing and outlandish, funny and serious, unexpected and curious. The one who looks sees. After all, even on a summer starry night, people will not notice ... radiance. The North Star, if they do not direct their gaze to a cloudless height.
Moving from class to class, I got acquainted with natural, fractional, decimal, negative, rational. This year I studied irrational. Among the irrational numbers there is a special number, the exact calculations of which have been carried out by scientists for many centuries. I met it back in the 6th grade while studying the topic “Circumference and area of a circle”. Attention was focused on the fact that quite often we will meet with him in the lessons in the senior classes. Practical tasks for finding the numerical value of the number π were interesting. The number π is one of the most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. Many interesting facts are connected with the number π, so it is of interest to 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 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, in front of me I put goal: explore the history of the number π and the significance of the number π at the present stage of development of mathematics.
Tasks:
Study the literature in order to obtain information about the history of the number π;
Establish some facts from the "modern biography" of the number π;
Practical calculation of the approximate value of the ratio of the circumference of a circle to its diameter.
Object of study:
Object of study: The number of PI.
Subject of study: Interesting facts related to the number PI.
2. The main part. The 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 that are used in mathematics, in the natural sciences, in engineering, and in everyday life, few numbers receive as much attention as the number pi. One book says, “Pi is capturing the minds of scientific geniuses and amateur mathematicians all over the world” (“Fractals for the Classroom”).
It can be found in probability theory, in solving problems with complex numbers, and in other areas of mathematics that are unexpected and far from geometry. The English mathematician August de Morgan once called "pi" "... the mysterious number 3.14159... that climbs through the door, through the window and through the roof." This mysterious number, associated with one of the three classic problems of Antiquity - the construction of a square whose area is equal to the area of a given circle - entails a trail of dramatic historical and curious entertaining facts.
Some even consider it one of the five most important numbers in mathematics. But, as the book Fractals for the Classroom notes, for all the importance of pi, “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 of the Greek alphabet "pi".
Numerically, π begins as 3.141592 and has an infinite mathematical duration.
4. The history of the number "pi"
According to experts, this number was discovered by the Babylonian Magi. It was used in the construction of the famous Tower of Babel. However, 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 the number pi, which expresses the ratio of the circumference of a circle to its diameter, began in ancient Egypt. Area of circle diameter d Egyptian mathematicians defined as (d-d/9) 2 (this notation 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 the oldest religions that 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 a fraction 3,162... Ancient Greeks Eudoxus, Hippocrates and other measurements of the circle were reduced to the construction of a segment, and the measurement of the circle - to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples have tried to express the ratio of the circumference of a circle to its diameter by a rational number.
Archimedes in the 3rd century BC. substantiated in his short work "Measurement of the circle" three positions:
Any circle is equal in size to a right triangle, the legs of which are respectively equal to the circumference 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 more 3 10/71 .
According to precise calculations Archimedes the ratio of circumference to diameter is 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 of this number was found: 3,1415927...
In the first half of the XV century. observatories Ulugbek, near Samarkand, astronomer and mathematician al-Kashi calculated pi with 16 decimal places. Al-Kashi made unique calculations that were needed to compile a table of sines with a step of 1" . These tables have played an important role in astronomy.
Half a century later in Europe F.Viet found pi with only 9 correct decimal places by doing 16 doublings of the number of polygon sides. But at the same time F.Viet was the first to notice that pi can be found using the limits of some series. This discovery was of great
value, as it allowed us to calculate pi with any accuracy. Only 250 years later al-Kashi his result was surpassed.
The 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 an approximate value of the number of PI.
There is also an alternative version of the holiday - July 22. It's called "Approximate Pi Day". The fact is that the representation of 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 drew attention to the fact that the date and time coincide with the first digits of the number π.
Interesting facts related to the number “”
Scientists at the University of Tokyo, led by Professor Yasumasa Canada, managed to set a world record in calculating the number pi up to 12411 trillion signs. For 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 whole palace of Castel del Monte, in the proportions of which PI can be calculated. Now the magical palace is under the protection of UNESCO.
How to remember the first digits of the number "".
The first three digits of the number \u003d 3.14 ... are not difficult to remember at all. And to remember more signs, there are funny sayings and poems. For example, these:
You just need 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 digits of the number :
In the following phrases, the signs of the number 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 in the number known behind the number the number, how to notice good luck ”
(3,14159265359)
5. The notation of the number pi
The first to introduce the notation for the ratio of the circumference of a circle to its diameter with the modern symbol pi was an English mathematician W. Johnson in 1706. As a symbol, he took the first letter of the Greek word "periphery", which means in translation "circle". Introduced W. Johnson the designation became common 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 XVIII century. A.M. Lazhandre based on works I.G. Lambert proved that pi is irrational. Then the German mathematician F. Lindeman based on research Sh. Ermita, found a rigorous proof that this number is not only irrational, but also transcendental, i.e. cannot be the root of an algebraic equation. The search for an exact expression for pi continued after the work F. Vieta. At the beginning of the XVII century. Dutch mathematician from Cologne Ludolf van Zeulen(1540-1610) (some historians call him L. van Keulen) found 32 correct signs. Since then (publication year 1615), the value of the number p with 32 decimal places has been called the number Ludolf.
6. How to remember the number "Pi" with an accuracy of up to eleven digits
The number "Pi" is the ratio of the circumference of a circle to its diameter, it is expressed as an infinite decimal fraction. 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 and 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 of alternating current". To memorize 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 according to Kiselev's geometry textbook, where it was given).
The couplet is written according to the rules of the old Russian spelling, according to which, after consonant must be placed at the end of a word "soft" or "solid" sign. Here it is, this wonderful historical couplet:
Who is joking and wishing soon
"Pi" to find out the number - already knows.
For those who are going to do accurate calculations in the future, it makes sense to remember this. So what is the number "Pi" with an accuracy of up to eleven digits? Count the number of letters in each word and write these numbers in a row (separate the first digit with a comma).
Such accuracy is already quite enough for engineering calculations. In addition to the old one, there is also a modern way of remembering, which was pointed out by a reader who identified himself as George:
So that we don't make mistakes
Must read correctly:
Three, fourteen, fifteen
Ninety-two and six.
We 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 keep repeating:
"Three, fourteen, fifteen,
Nine, twenty-six and five."
Well, mathematicians with the help of modern computers can calculate almost any number of digits of the number "Pi".
7. Record memorization of the number pi
Mankind has been trying to remember the signs of pi for a long time. But how to store infinity in memory? Favorite question of professional mnemonists. Many unique theories and techniques for mastering a huge amount of information have been developed. Many of them are tested on pi.
The world record set in the last century in Germany is 40,000 characters. On December 1, 2003, Alexander Belyaev set the Russian record for the values of pi in Chelyabinsk. In an hour and a half, with short breaks, Alexander wrote 2,500 digits of pi on the blackboard.
Before that, it was considered a record in Russia to list 2000 characters, which was done 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 periodically train.
Conclusion.
The number pi appears in formulas used in many fields. Physics, electrical engineering, electronics, probability theory, construction and navigation are just some of them. And it seems that just as there is no end to the signs of pi, so there is no end to the possibilities of 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 a large number of different formulas.
This and other interdependencies allowed mathematicians to further understand the nature of the number pi.
The exact value of the number π in the modern world is not only of its own scientific value, but is also used for very precise calculations (for example, the orbit of a satellite, the construction of giant bridges), as well as assessing the speed and power of modern computers.
At present, the number π is associated with an incomprehensible set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this indicates a growing interest in the most important mathematical constant, the study of which has been going on for more than twenty-two centuries.
The work I did was interesting. I wanted to learn about the history of the number pi, its practical application, and I think I have achieved my goal. Summing up the work, I come to the conclusion that this topic is relevant. Many interesting facts are connected with the number π, so it is of interest to study. In my work, I became more familiar with the number - one of the eternal values that mankind has been using for many centuries. Learned some aspects of its rich history. Found out why the ancient world did not know the correct ratio of circumference to diameter. I looked clearly in what ways you can get a number. Based on experiments, I calculated the approximate value of the number in various ways. Conducted processing and analysis of the results of the experiment.
Any student today should know what the number means and what the number is approximately equal to. After all, everyone has their first acquaintance with a number, using it when calculating the circumference, 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 circumference of a circle to its diameter is the same for all circles, but even with difficulty remember the numerical value of the number equal to 3 ,fourteen.
I tried to lift the veil of the rich history of the number, which mankind has been using for many centuries. I made a presentation for my work.
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 at school IV-VI grades. - M.: Enlightenment, 1982.
2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook - M .: Education, 1989.
3. Zhukov A.V. The ubiquitous number "pi". - M.: Editorial URSS, 2004.
4. Kympan F. The history of the number "pi". - M.: Nauka, 1971.
5. Svechnikov A.A. journey into the history of mathematics - M .: Pedagogy - 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/
