Calculation of the Nth digit of Pi without calculating the previous ones. This is the magic number pi

Studying Pi numbers begins in the elementary grades when students learn about the circle, circumference, and the value of Pi. Since the value of Pi is a constant meaning the ratio of the length of the circle itself to the length of the diameter of a given circle. For example, if we take a circle whose diameter is equal to one, then its length is equal to Pi number. This value of Pi is infinite in mathematical continuation, but there is also a generally accepted designation. It comes from a simplified spelling of the value of Pi, it looks like 3.14.

The Historical Birth of Pi

The number Pi supposedly got its roots in Ancient Egypt. Since ancient Egyptian scientists calculated the area of a circle using diameter D, which took the value D - D/92. Which corresponded to 16/92, or 256/81, which means Pi is 3.160.
India in the sixth century BC also touched on the number Pi, in the religion of Jainism, records were found that stated that the number Pi is equal to 10 in the square root, which means 3.162.

Archimedes' teachings on the measurement of the circle in the third century BC led him to the following conclusions:

Later, he substantiated his conclusions by a sequence of calculations using examples of correctly inscribed or described polygonal shapes with doubling the number of sides of these figures. In precise calculations, Archimedes concluded the ratio of diameter and circumference in numbers between 3 * 10/71 and 3 * 1/7, therefore the value of Pi is 3.1419... Since we have already talked about the infinite form of this value, it looks like 3, 1415927... And this is not the limit, because the mathematician Kashi in the fifteenth century calculated the value of Pi as a sixteen-digit value.
English mathematician Johnson W. in 1706, began to use the symbol pi for the symbol? (from Greek it is the first letter in the word circle).

Mysterious meaning.

The value of Pi is irrational and cannot be expressed in fraction form because fractions use whole values. It cannot be a root in the equation, which is why it also turns out to be transcendental; it is found by considering any processes, being refined due to the large number of considered steps of a given process. There have been many attempts to calculate the largest number of decimal places in Pi, which have resulted in tens of trillions of digits of a given decimal value.

Interesting fact: Oddly enough, the value of Pi has its own holiday. It is called International Pi Day. It is celebrated on March 14th. The date appeared thanks to the very value of Pi 3.14 (mm.yy) and the physicist Larry Shaw, who was the first to celebrate this holiday in 1987.

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March 14, 2012

On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: the numerical expression π (Pi) is 3.14 (3rd month (March) 14th).

For the first time, schoolchildren encounter this unusual number in the elementary grades when studying circles and circumferences. The number π is a mathematical constant that expresses the ratio of the circumference of a circle to the length of its diameter. That is, if you take a circle with a diameter equal to one, then the circumference will be equal to the number “Pi”. The number π has an infinite mathematical duration, but in everyday calculations a simplified spelling of the number is used, leaving only two decimal places - 3.14.

In 1987, this day was celebrated for the first time. Physicist Larry Shaw from San Francisco noticed that in the American date system (month/day), the date March 14 - 3/14 coincides with the number π (π = 3.1415926...). Typically celebrations begin at 1:59:26 pm (π = 3.14 15926 …).

History of Pi

It is assumed that the history of the number π begins in Ancient Egypt. Egyptian mathematicians determined the area of a circle with diameter D as (D-D/9) 2. From this entry it is clear that at that time the number π was equated to the fraction (16/9) 2, or 256/81, i.e. π 3.160...

In the VI century. BC. in India, in the religious book of Jainism, there are entries indicating that the number π at that time was taken equal to the square root of 10, which gives the fraction 3.162...
In the 3rd century. BC Archimedes in his short work “Measurement of a Circle” substantiated three propositions:

Every circle is equal in size to a right triangle, the legs of which are respectively equal to the length of the circle and its radius;
The areas of a circle are related to a square built on a diameter as 11 to 14;
The ratio of any circle to its diameter is less than 3 1/7 and greater than 3 10/71.

Archimedes justified the last position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of the circumference to the diameter is between the numbers 3 * 10 / 71 and 3 * 1/7, which means that the number “pi” is 3.1419... The true value of this ratio is 3.1415922653...
In the 5th century BC. Chinese mathematician Zu Chongzhi found a more accurate value for this number: 3.1415927...
In the first half of the 15th century. The astronomer and mathematician Kashi calculated π with 16 decimal places.

A century and a half later in Europe, F. Viet found the number π with only 9 regular decimal places: he made 16 doublings of the number of sides of polygons. F. Viet was the first to notice that π can be found using the limits of certain series. This discovery 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 tried to unravel the mystery of this mysterious number.

What is the difficulty in calculating the value of π?

The number π is irrational: it cannot be expressed as a fraction p/q, where p and q are integers; this number cannot be the root of an algebraic equation. It is impossible to specify an algebraic or differential equation whose root will be π, therefore this number is called transcendental and is calculated by considering a process and is 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 the sequence with an accuracy of 10 trillion digits after the decimal point.

The digits of the decimal representation of π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that this number contains all written and unwritten books in encrypted form; any information that can be imagined is found in the number π.

You can try to unravel the mystery of this number yourself. Of course, it will not be possible to write down the number “Pi” in full. But for the most curious, I suggest considering the first 1000 digits of the number π = 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Remember the number "Pi"

Currently, with the help of computer technology, ten trillion digits of the number “Pi” have been calculated. The maximum number of numbers that a person could remember is one hundred thousand.

To remember the maximum number of digits of the number “Pi”, various poetic “memories” are used, in which words with a certain number of letters are arranged in the same sequence as the numbers in the number “Pi”: 3.1415926535897932384626433832795…. To restore the number, you need to count the number of characters in each word and write it down in order.

So I know the number called “Pi”. Well done! (7 digits)

So Misha and Anyuta came running
They wanted to know the number Pi. (11 digits)

This I know and remember perfectly:
And many signs are unnecessary for me, in vain.
Let's trust our enormous knowledge
Those who counted the numbers of the armada. (21 digits)

Once at Kolya and Arina's
We ripped the feather beds.
The white fluff was flying and spinning,
Showered, froze,
Satisfied
He gave it to us
Old women's headache.
Wow, the spirit of fluff is dangerous! (25 characters)

You can use rhyming lines to help you remember the right number.

So that we don't make mistakes,
You need to read it correctly:
Ninety two and six

If you try really hard,
You can immediately read:
Three, fourteen, fifteen,
Ninety two and six.

Three, fourteen, fifteen,
Nine, two, six, five, three, five.
To do science,
Everyone should know this.

You can just try
And repeat more often:
"Three, fourteen, fifteen,
Nine, twenty-six and five."

Still have questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!

The history of the number Pi begins in Ancient Egypt and goes in parallel with the development of all mathematics. This is the first time we meet this quantity within the walls of the school.

The number Pi is perhaps the most mysterious of the infinite number of others. Poems are dedicated to him, artists depict him, and even a film was made about him. In our article we will look at the history of development and calculation, 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. It was originally called the Ludolph number, and it was proposed to be denoted by the letter Pi by the British mathematician Jones in 1706. After the work of Leonhard Euler in 1737, this designation became generally accepted.

Pi is an irrational number, meaning its value cannot be accurately expressed as a fraction m/n, where m and n are integers. This was first proven by Johann Lambert in 1761.

The history of the development of the number Pi goes back about 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 slightly more than three.

Archimedes proposed a mathematical method for calculating Pi, in which he inscribed regular polygons in a circle and described it 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 Chinese mathematician Zu Chongzhi in the 480s. He deduced that Pi ≈ 355/113 and showed that 3.1415926< Пи < 3,1415927.

Before 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. In his work “Treatise on the Circle,” he cited 17 digits of Pi, 16 of which turned out to be correct.

The Dutch mathematician Ludolf van Zeijlen reached 20 numbers in his calculations, devoting 10 years of his life to this. After his death, 15 more digits of Pi were discovered in his notes. He bequeathed that these numbers be 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 Fractals for the Classroom points out, as important as Pi is, “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 a few of them that are simply impossible to imagine without this mysterious number.

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Based on materials from the site Calculator888.ru - Pi number - meaning, history, who invented it.

pi number pi, pi fibonacci number
(listed in order of increasing accuracy)

Continued fraction

(This continued fraction is not periodic. Written in linear notation)

Trigonometry radian = 180°

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

The first 1000 decimal places of the number π This term has other meanings, see Pi. If we take the diameter of a circle as one, then the circumference is the number “pi” Pi in perspective

(pronounced "pi") is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter. Denoted by the letter "pi" of the Greek alphabet. Old name - Ludolph number.

1 Properties
- 1.1 Transcendence and irrationality
- 1.2 Relationships
2 History
- 2.1 Geometric period
- 2.2 Classic period
- 2.3 Era of computing
3 Rational approximations
4 Unsolved problems
5 Buffon's needle method
6 Mnemonic rules
7 Additional facts
8 culture
9 See also
10 Notes
11 Literature
12 Links

Properties

Transcendence and irrationality

- an irrational number, that is, its value cannot be accurately expressed as a fraction m/n, where m and n are integers. Therefore, its decimal representation never ends and is not periodic. The irrationality of a number was first proven by Johann Lambert in 1761 by decomposing the number into a continued fraction. In 1794, Legendre gave a more rigorous proof of the irrationality of numbers and.
- a transcendental number, that is, it cannot be the root of any polynomial with integer coefficients. The transcendence of number was proven in 1882 by Lindemann, a professor at the University of Königsberg and later at the University of Munich. The proof was simplified by Felix Klein in 1894.
- Since in Euclidean geometry the area of a circle and the circumference of a circle are functions of number, the proof of transcendence put an end to the dispute about the squaring of the circle, which lasted more than 2.5 thousand years.
In 1934 Gelfond proved the transcendence of number. In 1996, Yuri Nesterenko proved that for any natural number and are algebraically independent, which, in particular, implies the transcendence of the numbers and.
is an element of the period ring (and therefore a computable and arithmetic number). But it is unknown whether it belongs to the ring of periods.

Ratios

There are many number formulas:

Francois Viet:

Wallis formula:

Leibniz series:

Other rows:

Multiple rows:

Limits:

here are prime numbers

Euler's identity:

Other connections between constants:

T.n. "Poisson integral" or "Gauss integral"

Integral sine:

Expression via dilogarithm:

Through an improper integral

Story

Constant symbol

The British mathematician Jones first used the Greek letter designation for this number in 1706, and it became generally accepted after the work of Leonhard Euler in 1737.

This designation comes from the initial letter of the Greek words περιφέρεια - circle, periphery and περίμετρος - perimeter.

The history of numbers ran parallel to the development of all mathematics. Some authors divide the whole process into 3 periods: the ancient period, during which it was studied from the perspective of geometry, the classical era, which followed the development of mathematical analysis in Europe in the 17th century, and the era of digital computers.

Geometric period

The fact that the ratio of the circumference to the diameter is the same for any circle, and that this ratio is slightly more than 3, was known to ancient Egyptian, Babylonian, ancient Indian and ancient Greek geometers. The earliest known approximation dates back to 1900 BC. e.; these are 25/8 (Babylon) and 256/81 (Egypt), both values differ from the true value by no more than 1%. The Vedic text "Shatapatha-brahmana" gives as 339/108 ≈ 3.139.

Liu Hui's algorithm for computing

Archimedes may have been the first to propose a mathematical method of calculation. To do this, he inscribed regular polygons in a circle and described it around it. Taking the diameter of a circle to be one, Archimedes considered the perimeter of the inscribed polygon as a lower bound for the circumference of the circle, and the perimeter of the circumscribed polygon as an upper bound. Considering a regular 96-gon, Archimedes estimated and guessed that it was approximately equal to 22/7 ≈ 3.142857142857143.

Zhang Heng in the 2nd century clarified the meaning of the number, proposing two equivalents: 1) 92/29 ≈ 3.1724...; 2) ≈ 3.1622.

In India, Aryabhata and Bhaskara used the approximation 3.1416. Varahamihira in the 6th century uses approximation in the Pancha Siddhantika.

Around 265 AD e. mathematician Liu Hui from the Wei kingdom provided a simple and accurate iterative algorithm (English: Liu Hui's π algorithm) for calculations with any degree of accuracy. He independently carried out the calculation for the 3072-gon and obtained an approximate value for according to the following principle:

Liu Hui later came up with a quick calculation method and obtained an approximate value of 3.1416 with just a 96-gon, taking advantage of the fact that the difference in area of successive polygons forms a geometric progression with a denominator of 4.

In the 480s, Chinese mathematician Zu Chongzhi demonstrated that ≈ 355/113 and showed that 3.1415926< < 3,1415927, используя алгоритм Лю Хуэя применительно к 12288-угольнику. Это значение оставалось самым точным приближением числа в течение последующих 900 лет.

Classical period

Before the 2nd millennium, no more than 10 digits were known. Further major achievements in the study are associated with the development of mathematical analysis, in particular with the discovery of series that make it possible to calculate with any accuracy by summing the appropriate number of terms of the series. In the 1400s, Madhava of Sangamagrama found the first of these series:

This result is known as the Madhava-Leibniz series, or Gregory-Leibniz series (after it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century). However, this series converges to very slowly, which makes it difficult to calculate many digits of the number in practice - about 4000 terms of the series must be added to improve Archimedes' estimate. However, by transforming this series into

Madhava was able to calculate as 3.14159265359, correctly identifying 11 digits in the number's notation. This record was broken in 1424 by the Persian mathematician Jamshid al-Kashi, who in his work entitled “Treatise on the Circle” gave 17 digits of the number, of which 16 were correct.

The first major European contribution since Archimedes was that of the Dutch mathematician Ludolf van Zeijlen, who spent ten years calculating a number with 20 decimal digits (this result was published in 1596). Using Archimedes' method, he brought the doubling to an n-gon, where n = 60 229. Having outlined his results in the essay “On the Circle” (“Van den Circkel”), Ludolf ended it with the words: “Whoever has the desire, let him go further.” After his death, 15 more exact digits of the number were discovered in his manuscripts. Ludolf bequeathed that the signs he found be carved on his tombstone. In his honor, the number was sometimes called the “Ludolf number”, or “Ludolf constant”.

Around the same time, methods for analyzing and determining infinite series began to develop in Europe. The first such representation was Vieta's formula:

found by François Viète in 1593. Another famous result was the Wallis formula:

bred by John Wallis in 1655.

Similar works:

A product that proves its relationship with the Euler number e:

In modern times, analytical methods based on identities are used for calculations. The formulas listed above are of little use for computational purposes, since they either use slowly converging series or require a complex operation of extracting the square root.

The first effective formula was found in 1706 by John Machin.

Expanding the arctangent into a Taylor series

you can obtain a quickly convergent series suitable for calculating numbers with great accuracy.

Formulas of this type, now known as Machin-like formulas, were used to set several consecutive records and remained the best known methods for fast calculation in the computer age. An outstanding record was set by the phenomenal counter Johann Dase, who in 1844, at the behest of Gauss, used Machin's formula to calculate 200 digits in his head. The best result by the end of the 19th century was obtained by the Englishman William Shanks, who took 15 years to calculate 707 digits, although due to an error only the first 527 were correct. To avoid such errors, modern calculations of this kind are carried out twice. If the results match, then they are highly likely to be correct. Shanks' bug was discovered by one of the first computers in 1948; he counted 808 characters in a few hours.

Theoretical advances in the 18th century led to an understanding of the nature of number that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved irrationality in 1761, and Adrienne Marie Legendre proved irrationality in 1774. In 1735, a connection was established between prime numbers and when Leonhard Euler solved the famous Basel problem - the problem of finding the exact value

which makes up. Both Legendre and Euler suggested that it could be transcendental, which was eventually proven in 1882 by Ferdinand von Lindemann.

William Jones's A New Introduction to Mathematics from 1706 is believed to have been the first to introduce the Greek letter to represent this constant, but this notation became especially popular after Leonhard Euler adopted it in 1737. He wrote:

There are many other ways of finding the lengths or areas of the corresponding curve or plane figure, which can greatly facilitate practice; for example, in a circle, the diameter is related to the circumference as 1 to

See also: History of mathematical notation

Era of Computing

The era of digital technology in the 20th century led to an increase in the rate of emergence of computing records. John von Neumann and others used ENIAC in 1949 to calculate 2037 digits, which took 70 hours. Another thousand digits were obtained in subsequent decades, and the million mark was passed in 1973 (ten digits of the number is sufficient for all practical purposes). This progress has taken place not only due to faster hardware, but also thanks to algorithms. One of the most significant results was the discovery in 1960 of the fast Fourier transform, which made it possible to quickly perform arithmetic operations on very large numbers.

At the beginning of the 20th century, Indian mathematician Srinivasa Ramanujan discovered many new formulas for, some of which became famous because of their elegance and mathematical depth. One of these formulas is a series:

The Chudnovsky brothers found one similar to it in 1987:

which gives approximately 14 digits for each member of the series. The Chudnovskys used this formula to set several calculation records in the late 1980s, including one that produced 1,011,196,691 decimal expansion digits in 1989. This formula is used in programs that calculate on personal computers, as opposed to supercomputers that set modern records.

While a sequence typically improves accuracy by a fixed amount with each successive term, there are iterative algorithms that multiply the number of correct digits at each step, although at a high computational cost at each step. A breakthrough in this regard came in 1975, when Richard Brent and Eugene Salamin (mathematician) independently discovered the Gauss–Legendre algorithm, which, using only arithmetic, Each step doubles the number of known signs. The algorithm consists of setting initial values

and iterations:

until an and bn are close enough. Then the estimate is given by the formula

Using this scheme, 25 iterations are enough to produce 45 million decimal places. A similar algorithm that quadruples the accuracy at each step was found by Jonathan Borwein and Peter Borwein. Using these methods, Yasumasa Kanada and his group, starting in 1980, set most of the computing records, up to 206,158,430,000 characters in 1999. In 2002, Canada and his group set a new record of 1,241,100,000,000 decimal places. Although most previous Canadian records were set using the Brent-Salamin algorithm, the 2002 calculation used two Machin-type formulas that were slower but radically reduced memory usage. The calculation was performed on a 64-node Hitachi supercomputer with 1 terabyte of RAM, capable of 2 trillion operations per second.

An important recent development is the Bailey-Borwain-Plouffe formula, discovered in 1997 by Simon Plouffe and named after the authors of the paper in which it was first published. This formula

is notable in that it allows you to extract any specific hexadecimal or binary digit of a number without calculating the previous ones. From 1998 to 2000, the distributed project PiHex used a modified version of Fabrice Bellard's BBP formula to calculate the quadrillionth bit of a number that turned out to be zero.

In 2006, Simon Plouffe found a number of beautiful formulas using PSLQ. Let q = eπ, then

and other types

where q = eπ, k is an odd number, and a, b, c are rational numbers. If k is of the form 4m + 3, then this formula has a particularly simple form:

for rational p, whose denominator is a number that can be factorized well, although a rigorous proof has not yet been provided.

In August 2009, scientists from the Japanese University of Tsukuba calculated a sequence of 2,576,980,377,524 decimal places.

On December 31, 2009, French programmer Fabrice Bellard calculated a sequence of 2,699,999,990,000 decimal places on a personal computer.

On August 2, 2010, American student Alexander Yee and Japanese researcher Shigeru Kondo (Japanese) Russian. calculated the sequence with an accuracy of 5 trillion decimal places.

On October 19, 2011, Alexander Yee and Shigeru Kondo calculated the sequence to an accuracy of 10 trillion decimal places.

Rational approximations

- Archimedes (III century BC) - ancient Greek mathematician, physicist and engineer;

- Aryabhata (5th century AD) - Indian astronomer and mathematician;

- Zu Chongzhi (5th century AD) - Chinese astronomer and mathematician.

Comparison of approximation accuracy:

Unsolved problems

It is not known whether the numbers are algebraically independent.
The exact measure of irrationality for the numbers and is unknown (but it is known that for it does not exceed 7.6063).
The measure of irrationality is unknown for any of the following numbers: For none of them it is even known whether it is a rational number, an algebraic irrational number, or a transcendental number.
It is not known whether an integer is an integer for any positive integer (see tetration).
It is unknown whether it belongs to the period ring.
Until now, nothing is known about the normality of the number; it is not even known which of the digits 0-9 appear in the decimal representation of a number an infinite number of times.

Buffon's needle method

A needle is randomly thrown onto a plane lined with equidistant straight lines, the length of which is equal to the distance between adjacent straight lines, so that with each throw the needle either does not intersect the straight lines or intersects exactly one. It can be proven that the ratio of the number of intersections of the needle with any line to the total number of throws tends to as the number of throws increases to infinity. This needle method is based on probability theory and is the basis of the Monte Carlo method.

Mnemonic rules

Poems for memorizing 8-11 signs of the number π:

Memorization can be helped by observing poetic meter:

Three, fourteen, fifteen, nine two, six five, three five
Eight nine, seven and nine, three two, three eight, forty six
Two six four, three three eight, three two seven nine, five zero two
Eight eight and four, nineteen, seven, one

There are poems in which the first digits of the number π are encrypted as the number of letters in words:

Similar poems existed in pre-reform orthography. in the following poem, in order to find out the corresponding digit of the number π, you must also count the letter “er”:

Who will jokingly and soon wish
Find out, he already knows the number.

There are verses that make it easier to remember the number π in other languages. For example, this poem in French allows you to remember the first 126 digits of the number π.

Additional facts

Pi monument on the steps of the Seattle Museum of Art

The ancient Egyptians and Archimedes accepted values from 3 to 3.160, and Arab mathematicians calculated the number.
The world record for memorizing decimal places belongs to the Chinese Liu Chao, who in 2006 reproduced 67,890 decimal places without error within 24 hours and 4 minutes. In the same 2006, the Japanese Akira Haraguchi said that he remembered the number up to the 100-thousandth decimal place, but this could not be officially verified.
In the state of Indiana (USA), a bill was issued in 1897 (see: en:Indiana Pi Bill), which legally established the value of Pi equal to 3.2. This bill was prevented from becoming law due to the timely intervention of a Purdue University professor who was present in the state legislature during the consideration of this law.
"The number Pi for bowhead whales is three" is written in the 1960s Whaler's Handbook.
As of 2010, 5 trillion decimal places have been calculated.
As of 2011, 10 trillion decimal places have been calculated.
As of 2014, 13.3 trillion decimal places have been calculated.

In culture

There is a feature film named after the number Pi.
The unofficial holiday "Pi Day" is celebrated annually on March 14, which in American date format (month/day) is written as 3.14, which corresponds to the approximate value of the number. It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who noticed that on March 14 at exactly 01:59 the date and time coincided with the first digits of the number Pi = 3.14159.
Another date associated with the number is July 22, which is called “Pi Approximation Day”, since in the European date format this day is written as 22/7, and the value of this fraction is an approximation of the number .

Notes

This definition is only suitable for Euclidean geometry. In other geometries, the ratio of the circumference of a circle to the length of its diameter can be arbitrary. For example, in Lobachevsky geometry this ratio is less than
Lambert, Johann Heinrich. Mémoire sur quelques proprietés remarquables des quantités transcendentes circulaires et logarithmiques, pp. 265–322.
Klein's proof is appended to the work “Questions of Elementary and Higher Mathematics,” Part 1, published in Göttingen in 1908.
Weisstein, Eric W. The Gelfond Constant (English) on the Wolfram MathWorld website.
1 2 Weisstein, Eric W. Irrational number (English) on the Wolfram MathWorld website.
Modular functions and questions of transcendence
Weisstein, Eric W. Pi Squared (English) on the Wolfram MathWorld website.
Nowadays, with the help of a computer, the number is calculated with an accuracy of up to a million digits, which is of more technical than scientific interest, because in general no one needs such accuracy.
The accuracy of the calculation is usually limited by the available computer resources - most often time, somewhat less often - the amount of memory.
Brent, Richard (1975), Traub, J F, ed., "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", Analytic Computational Complexity (New York: Academic Press): 151–176, (English)
Jonathan M Borwein. Pi: A Source Book. - Springer, 2004. - ISBN 0387205713. (English)
1 2 David H. Bailey, Peter B. Borwein, Simon Plouffe. On the Rapid Computation of Various Polylogarithmic Constants // Mathematics of Computation. - 1997. - T. 66, issue. 218. - pp. 903-913. (English)
Fabrice Bellard. A new formula to compute the nth binary digit of pi (English). Retrieved January 11, 2010. Archived from the original on August 22, 2011.
Simon Plouffe. Indentities inspired by Ramanujan’s Notebooks (part 2) (English). Retrieved January 11, 2010. Archived from the original on August 22, 2011.
A new record for the accuracy of calculating the number π has been set
Pi Computation Record
The number "Pi" is calculated with record accuracy
1 2 5 Trillion Digits of Pi - New World Record
10 trillion digits of decimal expansion for π defined
1 2 Round 2…10 Trillion Digits of Pi
Weisstein, Eric W. The Measure of Irrationality (English) on the Wolfram MathWorld website.
Weisstein, Eric W. Pi (English) on the Wolfram MathWorld website.
en:Irrational number#Open questions
Some unsolved problems in number theory
Weisstein, Eric W. Transcendental number (English) on the Wolfram MathWorld website.
An introduction to irrationality and transcendence methods
Deception or delusion? Quantum No. 5 1983
G. A. Galperin. Billiard dynamic system for pi.
Ludolph's number. Pi. Pi.
Chinese student breaks Guiness record by reciting 67,890 digits of pi
Interview with Mr. Chao Lu
How can anyone remember 100,000 numbers? - The Japan Times, 12/17/2006.
Pi World Ranking List
The Indiana Pi Bill, 1897
V.I. Arnold likes to cite this fact, see for example the book What is Mathematics (ps), page 9.
Alexander J. Yee. y-cruncher - A Multi-Threaded Pi-Program. y-cruncher.
Los Angeles Times article "Would You Like a Piece"? (the name plays on the similarity in the spelling of the number and the word pie (English pie)) (inaccessible link since 05/22/2013 (859 days) - history, copy) (English).

Literature

Zhukov A.V. About the number π. - M.: MCMNO, 2002. - 32 p. - ISBN 5-94057-030-5.
Zhukov A.V. The ubiquitous number “pi”. - 2nd ed. - M.: LKI Publishing House, 2007. - 216 p. - ISBN 978-5-382-00174-6.
Perelman Ya. I. Quadrature of the circle. - L.: House of Entertaining Science, 1941.

Links

Weisstein, Eric W. Pi Formulas (English) on the Wolfram MathWorld website.
Different representations of Pi on Wolfram Alpha
sequence A000796 in OEIS

Numbers with proper names
Powers of a thousand	Thousand Million Billion Trillion Quadrillion… Centillion
Old Russian numbers	Darkness Legion (ignorant) Leodr Vran (raven) Deck
Other powers of ten	Myriad Googol Asankheya Googolplex
Powers of twelve	A Dozen Gross Mass
Other natural	Devil's Dozen Number of the Beast Ramanujan-Hardy Number Graham Number Skewes Number Moser Number
Other numbers	Pi Golden ratio Silver ratio e (Euler's number) Euler's constant - Mascheroni Feigenbaum's constants Gelfond's constant Brun's constant Catalan's constant Apéry's constant Imaginary unit

pi is the number of the beast, pi is the number of mach, pi is the number of pi, pi is the fibonacci number

Pi (number) Information About

What is Pi equal to? we know and remember from school. It is equal to 3.1415926 and so on... It is enough for an ordinary person 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 of mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you will notice many surprising things among the endless series of numbers. Is it possible that Pi is hiding the deepest secrets of the universe?

Infinite number

The number Pi itself appears in our world as the length of a circle whose diameter is equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi begins as 3.1415926 and goes to infinity in rows of numbers that are never repeated. The first surprising 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 the ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no equation (polynomial) with integer coefficients whose solution would be the number Pi.

The fact that the number Pi is transcendental was proved in 1882 by the German mathematician von Lindemann. It was this proof that became the answer to the question of whether it is possible, using a compass and a ruler, to draw a square whose area is equal to the area of a given circle. This problem is known as the search for squaring a circle, which has worried humanity since ancient times. It seemed that this problem had a simple solution and was about to be solved. But it was precisely the incomprehensible property of the number Pi that showed that there was no solution to the problem of squaring the circle.

For at least four and a half millennia, humanity has been trying to obtain an increasingly accurate value for Pi. For example, in the Bible in the Third Book of Kings (7:23), the number Pi is taken to be 3.

The Pi value of remarkable accuracy can be found in the Giza pyramids: 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 this ratio by accident. The same value was already obtained in relation to the calculation of the number Pi in the 3rd century BC by the great Archimedes.

In the Papyrus of Ahmes, 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 was equal to 3.1388...

For almost two thousand years after Archimedes, people tried to find ways to calculate Pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Marcus Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Aryabhata, 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 got more than 10 decimal places due to the complexity of the calculations.

Finally, in 1400, the Indian mathematician Madhava from Sangamagram calculated Pi with an accuracy of 13 digits (although he was still mistaken 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 it in his books - this became known after his death. Newton claimed that he calculated Pi purely out of boredom.

Around the same time, other lesser-known mathematicians also came forward and proposed new formulas for calculating the number Pi through trigonometric functions.

For example, this is the formula used to calculate Pi by astronomy teacher John Machin in 1706: PI / 4 = 4arctg(1/5) – arctg(1/239). Using analytical methods, Machin derived the number Pi to one hundred decimal places from this formula.

By the way, in the same 1706, the number Pi received an official designation in the form of a Greek letter: William Jones used it in his work on mathematics, taking the first letter of the Greek word “periphery,” which means “circle.” The great Leonhard Euler, born in 1707, popularized this designation, now known to any schoolchild.

Before the era of computers, mathematicians focused on calculating as many signs as possible. In this regard, sometimes funny things arose. Amateur mathematician W. Shanks calculated 707 digits of Pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoverys in Paris in 1937. However, nine years later, observant mathematicians discovered that only the first 527 characters were correctly calculated. The museum had to incur significant expenses to correct the error - now all the figures 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, was enormous in size 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 machine 70 hours.

As computers improved, our knowledge of Pi moved 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 surpassed the 10 trillion character mark.

Where else can you meet Pi?

So, often our knowledge about the number Pi remains at the school level, and we know for sure that this number is irreplaceable primarily in geometry.

In addition to formulas for the length and area of a circle, the number Pi is used in formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: in some places the formulas are simple and easy to remember, but in others 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 equal to 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 the series, Pi appears most unexpectedly in the famous Riemann zeta function. It’s impossible to talk about it in a nutshell, let’s just say that someday the number Pi will help find a formula for calculating prime numbers.

And absolutely surprisingly: Pi appears in two of the most beautiful “royal” formulas of mathematics - Stirling’s formula (which helps to find the approximate value of the factorial and gamma function) and Euler’s formula (which connects as many as five mathematical constants).

However, the most unexpected discovery awaited mathematicians in probability theory. The number Pi is also there.

For example, the probability that two numbers will be relatively prime is 6/PI^2.

Pi appears in Buffon's needle-throwing problem, formulated in the 18th century: what is the probability that a needle thrown onto a lined piece of paper 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 simply the ratio of circumference to diameter?

We can also meet Pi in physics. 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 appears in the arrangement of the electron orbitals of the hydrogen atom. And what is again most incredible is that the number Pi 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, on which the film of the same name is based, aliens tell 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 written.

This novel actually reflected a mystery that has occupied the minds of mathematicians all over the world: is Pi a normal number in which the digits are scattered with equal frequency, or is there something wrong with this number? And although scientists are inclined to the first option (but cannot prove it), the number Pi looks very mysterious. A Japanese man once calculated how many times the numbers 0 to 9 occur in the first trillion digits of Pi. And I saw that the numbers 2, 4 and 8 were more common than the others. This may be one of the hints that Pi is not entirely normal, and the numbers in it are indeed not random.

Let's remember everything we read above and ask ourselves, what other irrational and transcendental number is so often found in the real world?

And there are more 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 main character of the American TV series “Suspect,” Professor Finch, told students that due to the infinity of the number Pi, any combination of numbers can be found in it, ranging from the numbers of your date of birth to more complex numbers. For example, at position 762 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 at the 17,387,594,880th digit.

All this means that in the infinity of the number Pi one can find not only interesting combinations of numbers, but also the encoded text of “War and Peace”, the Bible and even the Main Secret of the Universe, if such exists.

By the way, about the Bible. The famous popularizer of mathematics, Martin Gardner, stated in 1966 that the millionth digit of 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 verse (3-14-16) the seventh word contains five letters. The millionth figure was reached eight years later. It was the number five.

Is it worth asserting after this that the number Pi is random?