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 gave way to 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 "... the 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, having 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 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. 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 in Old Testament, supposedly, messages to descendants are encoded, which 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, having discarded the twelfth digit in the number of PI after the point when calculating the length of the meridian, we will be mistaken by several millimeters. And when calculating the length of the Earth's orbit 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 number of PI considered today, which reflects the properties of the Universe, may 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 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.
One of the most mysterious numbers known to mankind, of course, is the number Π (read - pi). In algebra, this number reflects the ratio of the circumference of a circle to its diameter. Previously, this quantity was called the Ludolf number. How and where the number Pi came from is not known for certain, but mathematicians divide the entire history of the number Π into 3 stages, into the ancient, classical and era of digital computers.
The number P is irrational, that is, it cannot be represented as a simple fraction, where the numerator and denominator are integers. Therefore, such a number has no end and is periodic. For the first time, the irrationality of P was proved by I. Lambert in 1761.
In addition to this property, the number P cannot also be the root of any polynomial, and therefore is a number property, when it was proved in 1882, it put an end to the almost sacred dispute of mathematicians “about the squaring of the circle”, which lasted for 2,500 years.
It is known that the first to introduce the designation of this number was the Briton Jones in 1706. After Euler's work appeared, the use of such a designation became generally accepted.
To understand in detail what Pi is, it should be said that its use is so widespread that it is difficult to even name a field of science in which it would be dispensed with. One of the simplest and most familiar values from the school curriculum is the designation of the geometric period. The ratio of the length of a circle to the length of its diameter is constant and equal to 3.14. This value was known even to the most ancient mathematicians in India, Greece, Babylon, Egypt. The earliest version of calculating the ratio dates back to 1900 BC. e. A closer to the modern value of P was calculated by the Chinese scientist Liu Hui, in addition, he also invented a quick method for such a calculation. Its value remained generally accepted for almost 900 years.
The classical period in the development of mathematics was marked by the fact that in order to establish exactly what the number Pi is, scientists began to use the methods of mathematical analysis. In the 1400s, the Indian mathematician Madhava used the theory of series to calculate and determined the period of the number P with an accuracy of 11 digits after the decimal point. The first European, after Archimedes, who investigated the number P and made a significant contribution to its justification, was the Dutchman Ludolf van Zeulen, who already determined 15 digits after the decimal point, and wrote very entertaining words in his will: "... whoever is interested - let him go further." It was in honor of this scientist that the number P received its first and only nominal name in history.
The era of computer computing brought new details to the understanding of the essence of the number P. So, in order to find out what the number Pi is, in 1949 the ENIAC computer was used for the first time, one of the developers of which was the future "father" of the theory of modern computers J. The first measurement was carried out on for 70 hours and gave 2037 digits after the decimal point in the period of the number P. The mark of a million characters was reached in 1973. In addition, during this period, other formulas were established that reflect the number P. So, the Chudnovsky brothers were able to find one that made it possible to calculate 1,011,196,691 digits of the period.
In general, it should be noted that in order to answer the question: "What is the number Pi?", Many studies began to resemble competitions. Today, supercomputers are already dealing with the question of what it really is, the number Pi. Interesting Facts associated with these studies permeate almost the entire history of mathematics.
Today, for example, world championships in memorizing the number P are held and world records are set, the latter belongs to the Chinese Liu Chao, who named 67,890 characters in a little over a day. In the world there is even a holiday of the number P, which is celebrated as "Pi Day".
As of 2011, 10 trillion digits of the number period have already been established.
14 Mar 2012
On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: the numerical expression π (Pi) is 3.14 (3rd month (March) 14th day).
For the first time, schoolchildren come across this unusual number already in the elementary grades when studying a circle and a circle. The number π is a mathematical constant that expresses the ratio of the circumference of a circle to the length of its diameter. That is, if we take a circle with a diameter equal to one, then the circumference will be equal to the number "Pi". The number π has an infinite mathematical duration, but in everyday calculations they use a simplified spelling of the number, leaving only two decimal places, - 3.14.
In 1987 this day was celebrated for the first time. Physicist Larry Shaw from San Francisco noticed that in the American system of writing dates (month / day), the date March 14 - 3/14 coincides with the number π (π \u003d 3.1415926 ...). Celebrations usually start at 1:59:26 p.m. (π = 3.14 15926 …).
History of Pi
It is assumed that the history of the number π begins in ancient Egypt. Egyptian mathematicians determined the area of a circle with a diameter D as (D-D/9) 2 . From this entry it can be seen that at that time the number π was equated to the fraction (16/9) 2, or 256/81, i.e. π 3.160...
In the VI century. BC. in India, in the religious book of Jainism, there are records indicating that the number π at that time was taken equal to the square root of 10, which gives a fraction of 3.162 ...
In the III century. BC Archimedes in his short work "Measurement of the circle" substantiated 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 greater than 3 10/71.
Archimedes substantiated the latter position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons with doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of circumference to diameter is between 3*10/71 and 3*1/7, which means that the number "pi" is 3.1419... The true value of this ratio is 3.1415922653...
In the 5th century BC. Chinese mathematician Zu Chongzhi found a more accurate value for this number: 3.1415927...
In the first half of the XV century. astronomer and mathematician-Kashi calculated π with 16 decimal places.
A century and a half later, in Europe, F. Viet found the number π with only 9 correct decimal places: he made 16 doublings of the number of sides of polygons. F. Wiet was the first to notice that π can be found using the limits of some series. This discovery was of great importance, it made it possible to calculate π with any accuracy.
In 1706, the English mathematician W. Johnson introduced the notation for the ratio of the circumference of a circle to its diameter and designated it with the modern symbol π, the first letter of the Greek word periferia-circle.
For a long period of time, scientists around the world have been trying to unravel the mystery of this mysterious number.
What is the difficulty in calculating the value of π?
The number π is irrational: it cannot be expressed as a fraction p/q, where p and q are integers, this number cannot be the root of an algebraic equation. It is impossible to specify an algebraic or differential equation whose root is π, therefore this number is called transcendental and is calculated by considering a process and refined by increasing the steps of the process under consideration. Multiple attempts to calculate the maximum number of digits of the number π have led to the fact that today, thanks to modern computing technology, it is possible to calculate a sequence with an accuracy of 10 trillion digits after the decimal point.
The digits of the decimal representation of the number π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that in this number in encrypted form there are all written and unwritten books, any information that can only be represented is in the number π.
You can try to solve the mystery of this number yourself. Writing down the number "Pi" in full, of course, will not work. But I propose to the most curious to consider the first 1000 digits of the number π = 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
Remember the number "Pi"
Currently, with the help of computer technology, ten trillion digits of the number "Pi" has been calculated. The maximum number of digits that a person could remember is one hundred thousand.
To memorize the maximum number of characters of the number "Pi", various poetic "memory" are used, in which words with a certain number of letters are arranged in the same sequence as the numbers in the number "Pi": 3.1415926535897932384626433832795 .... To restore the number, you need to count the number of characters in each of the words and write it down in order.
So I know the number called "Pi". Well done! (7 digits)
So Misha and Anyuta came running
Pi to know the number they wanted. (11 digits)
This I know and remember very well:
Pi many signs are superfluous to me, in vain.
Let's trust the vast knowledge
Those who have counted, numbers armada. (21 digits)
Once at Kolya and Arina
We ripped the feather beds.
White fluff flew, circled,
Courageous, froze,
blissed out
He gave us
Headache of old women.
Wow, dangerous fluff spirit! (25 characters)
You can use rhyming lines that help you remember the right number.
So that we don't make mistakes
It needs to be read correctly:
ninety two and six
If you try hard
You can immediately read:
Three, fourteen, fifteen
Ninety-two and six.
Three, fourteen, fifteen
Nine, two, six, five, three, five.
To do science
Everyone should know this.
You can just try
And keep repeating:
"Three, fourteen, fifteen,
Nine, twenty-six and five."
Do you have any questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!
The history of the number Pi begins in ancient Egypt and goes in parallel with the development of all mathematics. We meet this value for the first time within the walls of the school.
The number Pi is perhaps the most mysterious of an infinite number of others. Poems are dedicated to him, artists portray him, and a film has even been made about him. In our article, we will look at the history of development and computing, as well as the areas of application of the Pi constant in our lives.
Pi is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter. Initially, it was called the Ludolf number, and it was proposed to denote it by the letter Pi by the British mathematician Jones in 1706. After the work of Leonhard Euler in 1737, this designation became generally accepted.
The number Pi is irrational, that is, its value cannot be expressed exactly as a fraction m/n, where m and n are integers. This was first proved by Johann Lambert in 1761.
The history of the development of the number Pi has already been around 4000 years. Even the ancient Egyptian and Babylonian mathematicians knew that the ratio of the circumference to the diameter is the same for any circle and its value is a little more than three.
Archimedes proposed a mathematical method for calculating Pi, in which he inscribed in a circle and described regular polygons around it. According to his calculations, Pi was approximately equal to 22/7 ≈ 3.142857142857143.
In the 2nd century, Zhang Heng proposed two values for pi: ≈ 3.1724 and ≈ 3.1622.
Indian mathematicians Aryabhata and Bhaskara found an approximate value of 3.1416.
The most accurate approximation of pi for 900 years was a calculation by the Chinese mathematician Zu Chongzhi in the 480s. He deduced that Pi ≈ 355/113 and showed that 3.1415926< Пи < 3,1415927.
Until the 2nd millennium, no more than 10 digits of Pi were calculated. Only with the development of mathematical analysis, and especially with the discovery of series, were subsequent major advances in the calculation of the constant made.
In the 1400s, Madhava was able to calculate Pi=3.14159265359. His record was broken by the Persian mathematician Al-Kashi in 1424. He in his work "Treatise on the Circumference" cited 17 digits of Pi, 16 of which turned out to be correct.
The Dutch mathematician Ludolf van Zeulen reached 20 numbers in his calculations, giving 10 years of his life for this. After his death, 15 more digits of pi were discovered in his notes. He bequeathed that these figures were carved on his tombstone.
With the advent of computers, the number Pi today has several trillion digits and this is not the limit. But, as noted in Fractals for the Classroom, for all the importance of pi, “it is difficult to find areas in scientific calculations that require more than twenty decimal places.”
In our life, the number Pi is used in many scientific fields. Physics, electronics, probability theory, chemistry, construction, navigation, pharmacology - these are just some of them that simply cannot be imagined without this mysterious number.
According to the site Calculator888.ru - Pi number - meaning, history, who invented it.
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 the inscribed and circumscribed polygons according to the method of Archimedes, he came up with \(60 \cdot 2^(29) \) - a square 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, Vieta, 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 substitution are either greater or less than number\(\pi \), and each time it gets 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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