Pi (number)

	Irrational numbers; ζ (3) - ρ - √ 2 - √ 3 - √ 5 - ln 2 - φ, Φ - ψ - α, δ - e - and π
Number system	Estimated number
Decimal	3.1415926535897932384626433832795 ...
Binary	11.00100100001111110110 ...
Hexadecimal	3.243F6A8885A308D31319 ...
Sixty	3; 08 29 44 00 47 25 53 07 ...
Rational approximations	22 ⁄ 7 , 179 ⁄ 57 , 223 ⁄ 71 , 333 ⁄ 106 , 355 ⁄ 113 , 103 993 ⁄ 33 102 (listed in order of increasing accuracy)
Continued fraction	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, ...] (This continued fraction is not periodic . Recorded in linear notation.)
Trigonometry	radian = 180 °

${\boldsymbol {\pi }}$ ${\ displaystyle {\ boldsymbol {\ pi}}}$ ${\ displaystyle {\ boldsymbol {\ pi}}}$ (pronounced “pi” ) is a mathematical constant equal to the ratio of the circumference of a circle to its diameter ^[2] . It is designated by the letter of the Greek alphabet " 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 4 999999 837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 598253 4904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

The first thousand decimal places of π ^[1]

If the diameter of the circle is unity, then the length of the circle is the number pi

Properties

Transcendence and Irrationality

$\pi$ ${\ displaystyle \ pi}$ - irrational number , that is, its value cannot be accurately expressed as a fraction ${\frac {m}{n}}$ ${\ displaystyle {\ frac {m} {n}}}$ where $m$ ${\ displaystyle m}$ and $n$ ${\ displaystyle n}$ - whole numbers. Therefore, its decimal representation never ends and is not periodic. Irrationality of number $\pi$ ${\ displaystyle \ pi}$ was first proved by Johann Lambert in 1761 ^[3] by decomposing the tangent into a continuous fraction . In 1794, Legendre provided stricter proof of the irrationality of numbers. $\pi$ ${\ displaystyle \ pi}$ and $\pi ^{2}$ ${\ displaystyle \ pi ^ {2}}$ .
$π$ {\ displaystyle \ pi} Is a transcendental number , that is, it cannot be the root of any polynomial with integer coefficients. Transcendence of number $π$ {\ displaystyle \ pi} It was proved in 1882 by a professor at Konigsberg , and later at the University of Munich, Lindemann . The proof was simplified by Felix Klein in 1894 ^[4] .
- Since in Euclidean geometry the area of a circle and the circumference are functions of a number $\pi$ ${\ displaystyle \ pi}$ then the proof of transcendence $\pi$ ${\ displaystyle \ pi}$ put an end to the debate about the quadrature of the circle , which lasted more than 2.5 thousand years.
In 1934, Gelfond proved the transcendence of the number $e^{\pi }$ ${\ displaystyle e ^ {\ pi}}$ ^[5] . In 1996, Yuri Nesterenko proved that for any natural $n$ ${\ displaystyle n}$ the numbers $\pi$ ${\ displaystyle \ pi}$ and $e^{\pi {\sqrt {n}}}$ ${\ displaystyle e ^ {\ pi {\ sqrt {n}}}}$ are algebraically independent , whence, in particular, follows the transcendence of numbers $\pi +e^{\pi },\pi e^{\pi }$ ${\ displaystyle \ pi + e ^ {\ pi}, \ pi e ^ {\ pi}}$ and $e^{\pi {\sqrt {n}}}$ ${\ displaystyle e ^ {\ pi {\ sqrt {n}}}}$ ^[6] ^[7] .
$\pi$ ${\ displaystyle \ pi}$ is an element of the ring of periods (and therefore a computable and arithmetic number ). But it is not known whether $1/\pi$ ${\ displaystyle 1 / \ pi}$ to the ring of periods.

Relationships

Many formulas are known for calculating the number $\pi$ ${\ displaystyle \ pi}$ :

The Vieta formula for approximating the number π :

{\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \ldots

{\ displaystyle {\ frac {2} {\ pi}} = {\ frac {\ sqrt {2}} {2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2}}}} { 2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2}}}}}} {2}} \ cdot \ ldots}

This is the first known explicit performance.

\pi

{\ displaystyle \ pi}

with an infinite number of operations. Applying identity

\sin 2\varphi =2\sin \varphi \cos \varphi

{\ displaystyle \ sin 2 \ varphi = 2 \ sin \ varphi \ cos \ varphi}

recursively and going to the limit, we get

\varphi \cos {\dfrac {\varphi }{2}}\cos {\frac {\varphi }{4}}\cdots =\sin \varphi .

{\ displaystyle \ varphi \ cos {\ dfrac {\ varphi} {2}} \ cos {\ frac {\ varphi} {4}} \ cdots = \ sin \ varphi.}

It remains to substitute

\varphi ={\frac {\pi }{2}}

{\ displaystyle \ varphi = {\ frac {\ pi} {2}}}

and use the double-angle cosine formula :

\cos 2\varphi =\cos ^{2}\varphi -\sin ^{2}\varphi .

{\ displaystyle \ cos 2 \ varphi = \ cos ^ {2} \ varphi - \ sin ^ {2} \ varphi.}

Wallis formula :

{\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {\pi }{2}}

{\ displaystyle {\ frac {2} {1}} \ cdot {\ frac {2} {3}} \ cdot {\ frac {4} {3}} \ cdot {\ frac {4} {5}} \ cdot {\ frac {6} {5}} \ cdot {\ frac {6} {7}} \ cdot {\ frac {8} {7}} \ cdot {\ frac {8} {9}} \ cdots = {\ frac {\ pi} {2}}}

Leibniz Row :

{\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}

{\ displaystyle {\ frac {1} {1}} - {\ frac {1} {3}} + {\ frac {1} {5}} - {\ frac {1} {7}} + {\ frac {1} {9}} - \ cdots = {\ frac {\ pi} {4}}}

Other ranks:

{\begin{aligned}\pi &={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {8}{8k+2}}+{\frac {4}{8k+3}}+{\frac {4}{8k+4}}-{\frac {1}{8k+7}}\right)=\\&={\frac {1}{4}}\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {8}{8k+1}}+{\frac {8}{8k+2}}+{\frac {4}{8k+3}}-{\frac {2}{8k+5}}-{\frac {2}{8k+6}}-{\frac {1}{8k+7}}\right)=\\&=\;\;\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k}}}\left({\frac {2}{4k+1}}+{\frac {2}{4k+2}}+{\frac {1}{4k+3}}\right)\end{aligned}}

{\ displaystyle {\ begin {aligned} \ pi & = {\ frac {1} {2}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {16 ^ {k}}} \ left ({\ frac {8} {8k + 2}} + {\ frac {4} {8k + 3}} + {\ frac {4} {8k + 4}} - {\ frac {1} {8k +7}} \ right) = \\ & = {\ frac {1} {4}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {16 ^ {k}}} \ left ({\ frac {8} {8k + 1}} + {\ frac {8} {8k + 2}} + {\ frac {4} {8k + 3}} - {\ frac {2} {8k + 5}} - {\ frac {2} {8k + 6}} - {\ frac {1} {8k + 7}} \ right) = \\ & = \; \; \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {4 ^ {k}}} \ left ({\ frac {2} {4k + 1}} + {\ frac {2} {4k + 2}} + {\ frac {1} {4k + 3}} \ right) \ end {aligned}}}

\pi =2{\sqrt {3}}\sum \limits _{k=0}^{\infty }{\frac {(-1)^{k}}{\,3^{k}\,(2k+1)}}

{\ displaystyle \ pi = 2 {\ sqrt {3}} \ sum \ limits _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {\, 3 ^ {k} \, (2k + 1)}}}

Multiple rows:

\pi =8\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {1}{(4m-2)^{2k}}}=4\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {m^{2}-k^{2}}{(m^{2}+k^{2})^{2}}}={\sqrt[{4\,\,}]{360\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{k}{\frac {1}{m(k+1)^{3}}}}}

{\ displaystyle \ pi = 8 \ sum \ limits _ {k = 1} ^ {\ infty} \ sum \ limits _ {m = 1} ^ {\ infty} {\ frac {1} {(4m-2) ^ {2k}}} = 4 \ sum \ limits _ {k = 1} ^ {\ infty} \ sum \ limits _ {m = 1} ^ {\ infty} {\ frac {m ^ {2} -k ^ { 2}} {(m ^ {2} + k ^ {2}) ^ {2}}} = {\ sqrt [{4 \, \,}] {360 \ sum \ limits _ {k = 1} ^ { \ infty} \ sum \ limits _ {m = 1} ^ {k} {\ frac {1} {m (k + 1) ^ {3}}}}}}

Limits :

\pi =\lim \limits _{m\rightarrow \infty }{\frac {(m!)^{4}\,{2}^{4m}}{\left[(2m)!\right]^{2}\,m}}

{\ displaystyle \ pi = \ lim \ limits _ {m \ rightarrow \ infty} {\ frac {(m!) ^ {4} \, {2} ^ {4m}} {\ left [(2m)! \ right ] ^ {2} \, m}}}

\pi ={\sqrt {\frac {6}{\lim \limits _{n\to \infty }\prod \limits _{k=1 \atop p_{k}\in \mathbf {P} }^{n}\,\left(1-{\frac {1}{p_{k}^{2}}}\right)}}}\quad \to

{\ displaystyle \ pi = {\ sqrt {\ frac {6} {\ lim \ limits _ {n \ to \ infty} \ prod \ limits _ {k = 1 \ atop p_ {k} \ in \ mathbf {P} } ^ {n} \, \ left (1 - {\ frac {1} {p_ {k} ^ {2}}} \ right)}}} \ quad \ to}

here

p_{k}

{\ displaystyle p_ {k}}

- prime numbers

\pi =\lim _{n\to \infty }2^{n}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}}}},

{\ displaystyle \ pi = \ lim _ {n \ to \ infty} 2 ^ {n} {\ sqrt {2 - {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2+ \ dots + {\ sqrt {2}}}}}}}}}},}

Where

n

{\ displaystyle n}

equal to the number of roots in the expression. ^[eight]

Euler Identity :

e^{i\pi }+1=0\;

{\ displaystyle e ^ {i \ pi} + 1 = 0 \;}

Other relationships between constants :

{\frac {\pi }{e}}=2\prod \limits _{k=1}^{\infty }\left({\frac {2k+1}{2k-1}}\right)^{2k-1}\left({\frac {k}{k+1}}\right)^{2k}

{\ displaystyle {\ frac {\ pi} {e}} = 2 \ prod \ limits _ {k = 1} ^ {\ infty} \ left ({\ frac {2k + 1} {2k-1}} \ right ) ^ {2k-1} \ left ({\ frac {k} {k + 1}} \ right) ^ {2k}}

\pi \cdot e=6\prod \limits _{k=1}^{\infty }\left({\frac {2k+3}{2k+1}}\right)^{2k+1}\left({\frac {k}{k+1}}\right)^{2k}

{\ displaystyle \ pi \ cdot e = 6 \ prod \ limits _ {k = 1} ^ {\ infty} \ left ({\ frac {2k + 3} {2k + 1}} \ right) ^ {2k + 1 } \ left ({\ frac {k} {k + 1}} \ right) ^ {2k}}

T. n. Poisson integral or Gauss integral

\int \limits _{-\infty }^{+\infty }\ e^{-x^{2}}{dx}={\sqrt {\pi }}

{\ displaystyle \ int \ limits _ {- \ infty} ^ {+ \ infty} \ e ^ {- x ^ {2}} {dx} = {\ sqrt {\ pi}}}

\int \limits _{0}^{\infty }{\frac {Br(x)}{\displaystyle e^{Br^{4}(x)}}}dx={\sqrt {\pi }},

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {Br (x)} {\ displaystyle e ^ {Br ^ {4} (x)}} dx = {\ sqrt {\ pi }},}

Where

Br(x)

{\ displaystyle Br (x)}

- Bring's root .

Integral Sine :

\int \limits _{-\infty }^{+\infty }{{\frac {\sin x}{x}}dx}=\pi

{\ displaystyle \ int \ limits _ {- \ infty} ^ {+ \ infty} {{\ frac {\ sin x} {x}} dx} = \ pi}

Expression through dilogarithm : ^[9]

\pi ={\sqrt {6\ln ^{2}2+12\ \operatorname {Li} _{2}\left({\frac {1}{2}}\right)}}

{\ displaystyle \ pi = {\ sqrt {6 \ ln ^ {2} 2 + 12 \ \ operatorname {Li} _ {2} \ left ({\ frac {1} {2}} \ right)}}}

Through an improper integral

\int \limits _{0}^{+\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi

{\ displaystyle \ int \ limits _ {0} ^ {+ \ infty} {\ frac {dx} {(x + 1) {\ sqrt {x}}}} = \ pi}

History

Constant symbol

For the first time the designation of this number in the Greek letter $\pi$ ${\ displaystyle \ pi}$ used the British mathematician Jones in 1706, ^[10] and it became generally accepted after the work of Leonard Euler in 1737.

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

Number history $\pi$ ${\ displaystyle \ pi}$ went in parallel with the development of all mathematics. Some authors divide the entire process into 3 periods: ancient period during which $\pi$ ${\ displaystyle \ pi}$ studied from the perspective of geometry , the classical era that 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 circumference to diameter is the same for any circumference, and that this ratio is slightly more than 3, was known even by ancient Egyptian , Babylonian , ancient Indian and ancient Greek geometers, the oldest approximations date back to the third millennium BC. e. In ancient Babylon took $\pi$ ${\ displaystyle \ pi}$ equal to three. Moreover, it was determined through the formula: the area of a circle is equal to the square of the circumference divided by 12. ^[11]

The earliest known approximations other than 3 date from approx. 1900 BC e.: it is 25/8 = 3.125 (clay tablet from the Suz of the period of the Old Babylonian kingdom ) ^[12] and ^256/81 ≈ 3.16 (Egyptian papyrus of Ahmes of the period of the Middle Kingdom ); both values differ from true by no more than 1%. The Vedic text of Satapatha Brahman gives $\pi$ ${\ displaystyle \ pi}$ as 339/108 ≈ 3.139.

Chinese philosopher and scientist Zhang Heng , in the 2nd century, proposed for the number $\pi$ ${\ displaystyle \ pi}$ two equivalents: 92/29 ≈ 3.1724 and ${\sqrt {10}}$ ${\ displaystyle {\ sqrt {10}}}$ ≈ 3.1622.

In the holy books of Jainism written in the 5th-6th centuries BC e., found then in India $\pi$ ${\ displaystyle \ pi}$ also accepted equal ${\sqrt {10}}$ ${\ displaystyle {\ sqrt {10}}}$ ^[13]

Liu Hui's algorithm for computing

\pi

{\ displaystyle \ pi}

Archimedes may have been the first to propose a mathematical method of computing $\pi$ ${\ displaystyle \ pi}$ . To do this, he inscribed in a circle and described regular polygons around it. Taking the diameter of the circle as unity, Archimedes considered the perimeter of the inscribed polygon as the lower bound for the circumference, and the perimeter of the described polygon as the upper bound. Considering the correct 96-gon, Archimedes received an estimate $3{\frac {10}{71}}<\pi <3{\frac {1}{7}}$ ${\ displaystyle 3 {\ frac {10} {71}} <\ pi <3 {\ frac {1} {7}}}$ and suggested for approximate calculation $\pi$ ${\ displaystyle \ pi}$ the upper of the boundaries found by him: - 22/7 ≈ 3.142857142857143. The next approximation in European culture is associated with the astronomer Claudius Ptolemy (c. 100 - c. 170), who created a table of chords with a step of half a degree, which allowed him to obtain for $\pi$ ${\ displaystyle \ pi}$ approximation ^377/120 , equal to approximately half of the perimeter of a 720-gon inscribed in it by a unit circle. ^[14] Leonardo of Pisa ( Fibonacci ) in the book Practica Geometriae (circa 1220), apparently taking the Ptolemy approximation as the lower bound for $\pi$ ${\ displaystyle \ pi}$ leads its approximation - ^864/275 . ^[15] But it turns out to be worse than that of Ptolemy, since the latter made a mistake when determining the chord length of half a degree up, as a result of which the approximation ^377/120 turned out to be the upper bound for $\pi$ ${\ displaystyle \ pi}$ .

In India, Ariabhata and Bhaskara used the approximation of 3.1416. In the VI century, Varahamihira uses the Panca Siddhantika approximation ${\sqrt {10}}$ ${\ displaystyle {\ sqrt {10}}}$ .

Around 265 BC e. mathematician Liu Hui from the Wei kingdom provided a simple and accurate iterative algorithm ( Liu Hui's π algorithm ) for computing $\pi$ ${\ displaystyle \ pi}$ with any degree of accuracy. He independently calculated for the 3072-gon and obtained an approximate value for $\pi$ ${\ displaystyle \ pi}$ according to the following principle:

\pi \approx A_{3072}={3\cdot 2^{8}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+1}}}}}}}}}}}}}}}}}}}\approx 3,14159.

{\ displaystyle \ pi \ approx A_ {3072} = {3 \ cdot 2 ^ {8} \ cdot {\ sqrt {2 - {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2 + 1}}}}}}}}}}}}}}}}}}}} approx 3 , 14159.}

Liu Hui later came up with a quick calculation method. $\pi$ ${\ displaystyle \ pi}$ and got an approximate value of 3.1416 with only a 96-gon, taking advantage of the fact that the difference in the area of the successive polygons forms a geometric progression with the denominator of 4.

In the 480s, Chinese mathematician Zu Chunzhi demonstrated that $\pi$ ${\ displaystyle \ pi}$ ≈ 355/113, and showed that 3.1415926 < $\pi$ ${\ displaystyle \ pi}$ <3.1415927 using the Liu Hui algorithm for the 12288-gon. This value remained the most accurate approximation of the number $\pi$ ${\ displaystyle \ pi}$ over the next 900 years.

Classic Period

Until the 2nd millennium, no more than 10 digits were known $\pi$ ${\ displaystyle \ pi}$ . Further major research achievements $\pi$ ${\ displaystyle \ pi}$ associated with the development of mathematical analysis , in particular with the opening of series to calculate $\pi$ ${\ displaystyle \ pi}$ with any accuracy, summing up the appropriate number of members of the series.

Madhava Series - Leibniz

In the 1400s, Madhava from Sangamagram found the first of these rows:

{\pi }={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+\cdots

{\ displaystyle {\ pi} = {\ frac {4} {1}} - {\ frac {4} {3}} + {\ frac {4} {5}} - {\ frac {4} {7} } + \ cdots}

This result is known as the Madhava-Leibniz series, or the Gregory-Leibniz series (after it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century). However, this series converges to $\pi$ ${\ displaystyle \ pi}$ very slowly, which leads to the difficulty of calculating many digits of a number in practice - it is necessary to add about 4000 members of the series in order to improve the estimate of Archimedes. However, converting this series to

\pi ={\sqrt {12}}\,\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{5\cdot 3^{2}}}-{\frac {1}{7\cdot 3^{3}}}+\cdots \right)

{\ displaystyle \ pi = {\ sqrt {12}} \, \ left (1 - {\ frac {1} {3 \ cdot 3}} + {\ frac {1} {5 \ cdot 3 ^ {2}} } - {\ frac {1} {7 \ cdot 3 ^ {3}}} + \ cdots \ right)}

Madhava was able to figure out $\pi$ ${\ displaystyle \ pi}$ as 3.14159265359, correctly identifying 11 digits in the record number. This record was broken in 1424 by the Persian mathematician Jamshid al-Kashi , who in his work under the title "Treatise on the Circumference" cited 17 digits of the number $\pi$ ${\ displaystyle \ pi}$ of which 16 are faithful.

Ludolph number

The first major European contribution since Archimedes was the contribution of the Dutch mathematician Ludolf van Zeilain , who spent ten years calculating the number $\pi$ ${\ displaystyle \ pi}$ with 20 decimal digits (this result was published in 1596). Using the method of Archimedes, he brought the doubling to an n- gon, where n = 60 · 2 ²⁹ . Having set forth his results in the essay “On the Circle” (“Van den Circkel”), Ludolf ended it with the words: “Whoever has a hunt, let him go further.” After death, another 15 exact digits of the number were found in his manuscripts. $\pi$ ${\ displaystyle \ pi}$ . Ludolf bequeathed that the signs he found were carved on his headstone. In honor of him the number $\pi$ ${\ displaystyle \ pi}$ sometimes called the “Ludolph number” or “Ludolph constant”.

Ludolph number is an approximate value for a number $\pi$ ${\ displaystyle \ pi}$ with 32 valid decimal places.

Vieta formula for approximation π

Around the same time, methods for analyzing and determining infinite series began to develop in Europe. The first such presentation was the Vieta formula for approximating the number π :

{\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots

{\ displaystyle {\ frac {2} {\ pi}} = {\ frac {\ sqrt {2}} {2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2}}}} { 2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2}}}}}} {2}} \ cdot \ cdots}

,

found by Francois Viet in 1593.

Wallis Formula

Another well-known result was the Wallis formula :

{\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots

{\ displaystyle {\ frac {\ pi} {2}} = {\ frac {2} {1}} \ cdot {\ frac {2} {3}} \ cdot {\ frac {4} {3}} \ cdot {\ frac {4} {5}} \ cdot {\ frac {6} {5}} \ cdot {\ frac {6} {7}} \ cdot {\ frac {8} {7}} \ cdot { \ frac {8} {9}} \ cdots}

,

Bred by John Wallis in 1655.

Similar works:

$\pi =3\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {1}{3}}\right)^{2}}}$ ${\ displaystyle \ pi = 3 \ cdot {\ frac {\ sqrt {3}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2}} {k ^ {2} - \ left ({\ frac {1} {3}} \ right) ^ {2}}}}$

$\pi ={\frac {3}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {2}{3}}\right)^{2}}}$ ${\ displaystyle \ pi = {\ frac {3} {2}} \ cdot {\ frac {\ sqrt {3}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2}} {k ^ {2} - \ left ({\ frac {2} {3}} \ right) ^ {2}}}}$

$\pi =4\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {1}{4}}\right)^{2}}}$ ${\ displaystyle \ pi = 4 \ cdot {\ frac {\ sqrt {2}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2}} {k ^ {2} - \ left ({\ frac {1} {4}} \ right) ^ {2}}}}$

$\pi ={\frac {4}{3}}\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {3}{4}}\right)^{2}}}$ ${\ displaystyle \ pi = {\ frac {4} {3}} \ cdot {\ frac {\ sqrt {2}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2}} {k ^ {2} - \ left ({\ frac {3} {4}} \ right) ^ {2}}}}$

$\pi =6\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {1}{6}}\right)^{2}}}$ ${\ displaystyle \ pi = 6 \ cdot {\ frac {1} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2}} {k ^ {2} - \ left ({\ frac {1} {6}} \ right) ^ {2}}}}$

$\pi ={\frac {6}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {5}{6}}\right)^{2}}}$ ${\ displaystyle \ pi = {\ frac {6} {5}} \ cdot {\ frac {1} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ { 2}} {k ^ {2} - \ left ({\ frac {5} {6}} \ right) ^ {2}}}}$

$\pi =4\cdot \prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {1}{4}}}}$ ${\ displaystyle \ pi = 4 \ cdot \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2} + k} {k ^ {2} + k + {\ frac {1} {four}}}}}$

$\pi ={\frac {9}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {2}{9}}}}$ ${\ displaystyle \ pi = {\ frac {9} {2}} \ cdot {\ frac {\ sqrt {3}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2} + k} {k ^ {2} + k + {\ frac {2} {9}}}}}$

$\pi ={\frac {16}{3}}\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {3}{16}}}}$ ${\ displaystyle \ pi = {\ frac {16} {3}} \ cdot {\ frac {\ sqrt {2}} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ {2} + k} {k ^ {2} + k + {\ frac {3} {16}}}}}$

$\pi ={\frac {36}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {5}{36}}}}$ ${\ displaystyle \ pi = {\ frac {36} {5}} \ cdot {\ frac {1} {2}} \ prod \ limits _ {k = 1} ^ {\ infty} {\ frac {k ^ { 2} + k} {k ^ {2} + k + {\ frac {5} {36}}}}}$

Произведение, доказывающее родственную связь с числом Эйлера e

$\pi =2{\sqrt {3}}\prod \limits _{k=1}^{\infty }{\frac {\left(2k-1\right)^{{\frac {1}{2}}-k}\left(2k+3\right)^{k+{\frac {1}{2}}}}{2k+1}}\left({\frac {k}{k+1}}\right)^{2k}$ $\pi =2{\sqrt {3}}\prod \limits _{k=1}^{\infty }{\frac {\left(2k-1\right)^{{\frac {1}{2}}-k}\left(2k+3\right)^{k+{\frac {1}{2}}}}{2k+1}}\left({\frac {k}{k+1}}\right)^{2k}$

Методы, основанные на тождествах

В Новое время для вычисления $\pi$ $\pi$ используются аналитические методы, основанные на тождествах. Перечисленные выше формулы малопригодны для вычислительных целей, поскольку либо используют медленно сходящиеся ряды, либо требуют сложной операции извлечения квадратного корня.

Формулы Мэчина

Первый эффективный и современный способ нахождения числа π (а также натуральных логарифмов и других функций), основанный на развитой им теории рядов и математического анализа, дал в 1676 году Исаак Ньютон во втором письме к Ольденбургу ^[16] , разлагая в ряд $\mathrm {arctg} {\frac {1}{2}}$ $\mathrm {arctg} {\frac {1}{2}}$ . Based on this method, John Machin found the most effective formula in 1706 .

{\frac {\pi }{4}}=4\,\mathrm {arctg} {\frac {1}{5}}-\mathrm {arctg} {\frac {1}{239}}

{\ displaystyle {\ frac {\ pi} {4}} = 4 \, \ mathrm {arctg} {\ frac {1} {5}} - \ mathrm {arctg} {\ frac {1} {239}}}

Expanding Arctangent in a Taylor Series

\mathrm {arctg} \ x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots

{\ displaystyle \ mathrm {arctg} \ x = x - {\ frac {x ^ {3}} {3}} + {\ frac {x ^ {5}} {5}} - {\ frac {x ^ { 7}} {7}} + \ cdots}

,

one can obtain a rapidly converging series suitable for calculating the number $\pi$ ${\ displaystyle \ pi}$ with great accuracy.

Formulas of this type, currently known as Machin-like formulas , were used to set several consecutive records and remained the best known methods for quick calculation. $\pi$ ${\ displaystyle \ pi}$ in the era of computers. An outstanding record was set by the phenomenal counter Johann Dase , who in 1844, by order of Gauss, applied Machin's formula to calculate 200 digits $\pi$ ${\ displaystyle \ pi}$ . 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 likely to be true. Shanks's mistake was discovered by one of the first computers in 1948; in a few hours he counted 808 characters $\pi$ ${\ displaystyle \ pi}$ .

Pi - Transcendental Number

Theoretical achievements in the 18th century led to a comprehension of the nature of numbers $\pi$ ${\ displaystyle \ pi}$ , which could not be achieved only with the help of one numerical calculation. Johann Heinrich Lambert proved irrationality $\pi$ ${\ displaystyle \ pi}$ in 1761, and Adrien Marie Legendre in 1774 proved irrationality $\pi ^{2}$ ${\ displaystyle \ pi ^ {2}}$ . In 1735, a connection was established between primes and $\pi$ ${\ displaystyle \ pi}$ when Leonard Euler solved the famous Basel problem - the problem of finding the exact value

{\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots

{\ displaystyle {\ frac {1} {1 ^ {2}}} + {\ frac {1} {2 ^ {2}}} + {\ frac {1} {3 ^ {2}}} + {\ frac {1} {4 ^ {2}}} + \ cdots}

,

which turned out to be equal ${\frac {\pi ^{2}}{6}}$ ${\ displaystyle {\ frac {\ pi ^ {2}} {6}}}$ . Both Legendre and Euler assumed that $\pi$ ${\ displaystyle \ pi}$ may be transcendental , which was eventually proved in 1882 by Ferdinand von Lindeman .

Symbol " $\pi$ ${\ displaystyle \ pi}$ "

It is believed that the book of William Jones "A New Introduction to Mathematics" c 1706 the first introduced the Greek letter $\pi$ ${\ displaystyle \ pi}$ to denote this constant, but this entry became especially popular after Leonard Euler adopted it in 1737. He wrote: There are many other ways to find the lengths or areas of the corresponding curve or flat figure, which can greatly facilitate the practice; for example, in a circle, the diameter refers to the circumference as 1 to $\left({\frac {16}{5}}-{\frac {4}{239}}\right)-{\frac {1}{3}}\cdot \left({\frac {16}{5^{3}}}-{\frac {4}{239^{3}}}\right)+\cdots =3{,}14159\cdots =\pi$ ${\ displaystyle \ left ({\ frac {16} {5}} - {\ frac {4} {239}} \ right) - {\ frac {1} {3}} \ cdot \ left ({\ frac { 16} {5 ^ {3}}} - {\ frac {4} {239 ^ {3}}} \ right) + \ cdots = 3 {,} 14159 \ cdots = \ pi}$

The era of computer computing

The era of digital technology in the 20th century has led to an increase in the speed of computational records. John von Neumann and others used ENIAC in 1949 to calculate 2037 digits $\pi$ ${\ displaystyle \ pi}$ which took 70 hours. `In 1961, Daniel Shanks on the IBM 7090 calculated 100,000 characters, and a million mark was passed in 1973 ^[17] . Such progress has occurred not only due to faster hardware, but also thanks to new algorithms.

At the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan discovered many new formulas for $\pi$ ${\ displaystyle \ pi}$ some of which have become famous because of their elegance and mathematical depth. One of these formulas is a series:

{\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}

{\ displaystyle {\ frac {1} {\ pi}} = {\ frac {2 {\ sqrt {2}}} {9801}} \ sum _ {k = 0} ^ {\ infty} {\ frac {( 4k)! (1103 + 26390k)} {(k!) ^ {4} 396 ^ {4k}}}}

.

Brothers Chudnovsky in 1987 found similar to her:

{\frac {1}{\pi }}={\frac {1}{426880{\sqrt {10005}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}}

{\ displaystyle {\ frac {1} {\ pi}} = {\ frac {1} {426880 {\ sqrt {10005}}}} \ sum _ {k = 0} ^ {\ infty} {\ frac {( 6k)! (13591409 + 545140134k)} {(3k)! (K!) ^ {3} (- 640320) ^ {3k}}}}

,

which gives about 14 digits for each member of the series. Chudnovsky used this formula to set several records in the calculation $\pi$ ${\ displaystyle \ pi}$ in the late 1980s, including one that resulted in 1,011,196,691 decimal digits in 1989. This formula is used in programs that calculate $\pi$ ${\ displaystyle \ pi}$ on personal computers, unlike supercomputers , which set modern records.

While a sequence usually increases accuracy by a fixed amount with each subsequent term, there are iterative algorithms that at each step multiply the number of correct digits, requiring, however, high computational costs at each of these steps. A breakthrough in this regard was made in 1975, when Richard Brent and Eugene Salamin ( English Eugene Salamin (mathematician) ) independently discovered the Brent-Salamin algorithm ( English Gauss – Legendre algorithm ), which, using only arithmetic, doubles the number of known characters at each step ^[18] . The algorithm consists of setting the initial values

a_{0}=1\quad \quad \quad b_{0}={\frac {1}{\sqrt {2}}}\quad \quad \quad t_{0}={\frac {1}{4}}\quad \quad \quad p_{0}=1

{\ displaystyle a_ {0} = 1 \ quad \ quad \ quad b_ {0} = {\ frac {1} {\ sqrt {2}}} \ quad \ quad \ quad t_ {0} = {\ frac {1 } {4}} \ quad \ quad \ quad p_ {0} = 1}

and iterations:

a_{n+1}={\frac {a_{n}+b_{n}}{2}}\quad \quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}}

{\ displaystyle a_ {n + 1} = {\ frac {a_ {n} + b_ {n}} {2}} \ quad \ quad \ quad b_ {n + 1} = {\ sqrt {a_ {n} b_ {n}}}}

t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2}\quad \quad \quad p_{n+1}=2p_{n}

{\ displaystyle t_ {n + 1} = t_ {n} -p_ {n} (a_ {n} -a_ {n + 1}) ^ {2} \ quad \ quad \ quad p_ {n + 1} = 2p_ {n}}

,

until a _n and b _n become close enough. Then the score $\pi$ ${\ displaystyle \ pi}$ is given by the formula

\pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.

{\ displaystyle \ pi \ approx {\ frac {(a_ {n} + b_ {n}) ^ {2}} {4t_ {n}}}.}

Using this scheme, 25 iterations are enough to get 45 million decimal places. A similar algorithm that quadruples accuracy at each step was found by Jonathan Borwein, Peter Borwein ^[19] . Using these methods, Yasumasa Canada and his group, since 1980, set the majority of calculation records $\pi$ ${\ displaystyle \ pi}$ up to 206 158 430 000 characters in 1999. In 2002, Canada and its group set a new record - 1,241.1 billion decimal places. Although most of Canada’s previous records were set using the Brent-Salamin algorithm, the 2002 calculation used two formulas like the Mechin formulas, which worked more slowly but drastically reduced memory usage. The calculation was performed on a Hitachi supercomputer of 64 nodes with 1 terabyte of RAM, capable of performing 2 trillion operations per second.

An important development of recent times has become the Bailey – Borwain – Puff formula , discovered in 1997 by Simon Plouffe and named after the authors of the article in which it was first published ^[20] . This formula

\pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right),

{\ displaystyle \ pi = \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {16 ^ {k}}} \ left ({\ frac {4} {8k + 1}} - { \ frac {2} {8k + 4}} - {\ frac {1} {8k + 5}} - {\ frac {1} {8k + 6}} \ right),}

remarkable in that it allows you to extract any specific hexadecimal or binary digit of a number $\pi$ ${\ displaystyle \ pi}$ without calculating the previous ones ^[20] . From 1998 to 2000, the distributed PiHex project used the modified Fabs Bellar BBP formula to calculate the quadrillion bits of a number $\pi$ ${\ displaystyle \ pi}$ , which turned out to be zero ^[21] .

In 2006, Simon Pluff, using PSLQ, found a number of beautiful formulas ^[22] . Let q = e ^π ; then

{\frac {\pi }{24}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {3}{q^{n}-1}}-{\frac {4}{q^{2n}-1}}+{\frac {1}{q^{4n}-1}}\right)

{\ displaystyle {\ frac {\ pi} {24}} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n}} \ left ({\ frac {3} {q ^ {n} -1}} - {\ frac {4} {q ^ {2n} -1}} + {\ frac {1} {q ^ {4n} -1}} \ right)}

{\frac {\pi ^{3}}{180}}=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\left({\frac {4}{q^{n}-1}}-{\frac {5}{q^{2n}-1}}+{\frac {1}{q^{4n}-1}}\right)

{\ displaystyle {\ frac {\ pi ^ {3}} {180}} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {3}}} \ left ({ \ frac {4} {q ^ {n} -1}} - {\ frac {5} {q ^ {2n} -1}} + {\ frac {1} {q ^ {4n} -1}} \ right)}

and other types

\pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right)

{\ displaystyle \ pi ^ {k} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {k}}} \ left ({\ frac {a} {q ^ { n} -1}} + {\ frac {b} {q ^ {2n} -1}} + {\ frac {c} {q ^ {4n} -1}} \ right)}

,

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

p\pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {2^{k-1}}{q^{n}-1}}-{\frac {2^{k-1}+1}{q^{2n}-1}}+{\frac {1}{q^{4n}-1}}\right)

{\ displaystyle p \ pi ^ {k} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {k}}} \ left ({\ frac {2 ^ {k- 1}} {q ^ {n} -1}} - {\ frac {2 ^ {k-1} +1} {q ^ {2n} -1}} + {\ frac {1} {q ^ {4n } -1}} \ right)}

for rational p , whose denominator is a factorable number, although no rigorous proof has yet been provided.

In August 2009, scientists from the Tsukuba University of Japan calculated a sequence of 2 576 980 377 524 decimal places ^[23] .

On December 31, 2009, the French programmer Fabrice Bellard calculated a sequence of 2,699,999,990,000 decimal digits on a personal computer ^[24] .

August 2, 2010 American student Alexander Yi and Japanese researcher Shigeru Kondo the sequence was calculated with an accuracy of 5 trillion digits after the decimal point ^[25] ^[26] .

On October 19, 2011, Alexander Yi and Shigeru Kondo calculated the sequence with an accuracy of 10 trillion digits after the decimal point ^[27] ^[28] .

The Dutch mathematician Brouwer in the first half of the 20th century cited the search in decimal expansion as an example of a meaningless task $\pi$ ${\ displaystyle \ pi}$ sequences $0123456789$ ${\ displaystyle 0123456789}$ - in his opinion, the accuracy necessary for this will never be achieved. At the end of the 20th century, this sequence was discovered; it begins with the 17,387,594,880th decimal place ^[29] .

Rational approximations

${\frac {22}{7}}$ ${\ displaystyle {\ frac {22} {7}}}$ - Archimedes (III century BC) - an ancient Greek mathematician, physicist and engineer;
${\frac {377}{120}}$ ${\ displaystyle {\ frac {377} {120}}}$ - Claudius Ptolemy (II century A.D.) and Ariabhata (V century A.D.) - Indian astronomer and mathematician;
${\frac {355}{113}}$ ${\ displaystyle {\ frac {355} {113}}}$ - Zu Chunzhi (V century A.D. ) - Chinese astronomer and mathematician.

Comparison of Accuracy

Number	Rounded value	Accuracy (match bits )
$\pi$ ${\ displaystyle \ pi}$	3.14159265 ...
${\frac {22}{7}}$ ${\ displaystyle {\ frac {22} {7}}}$	3.14 285714 ...	2 decimal places
${\frac {377}{120}}$ ${\ displaystyle {\ frac {377} {120}}}$	3,141 66667 ...	3 decimal places
${\frac {355}{113}}$ ${\ displaystyle {\ frac {355} {113}}}$	3.141592 92 ...	6 decimal places

Open Issues

The exact measure of irrationality for numbers is unknown. $\pi$ ${\ displaystyle \ pi}$ and $\pi ^{2}$ ${\ displaystyle \ pi ^ {2}}$ (but it is known that for $\pi$ ${\ displaystyle \ pi}$ it does not exceed 7.6063) ^[30] ^[31] .
Unknown measure of irrationality for any of the following numbers: $\pi +e,\pi -e,\pi \cdot e,{\frac {\pi }{e}},\pi ^{e},\pi ^{\sqrt {2}},\ln \pi ,\pi ^{\pi },e^{\pi ^{2}}.$ ${\ displaystyle \ pi + e, \ pi -e, \ pi \ cdot e, {\ frac {\ pi} {e}}, \ pi ^ {e}, \ pi ^ {\ sqrt {2}}, \ ln \ pi, \ pi ^ {\ pi}, e ^ {\ pi ^ {2}}.}$ It is not even known to any of them whether it is a rational number, an algebraic irrational, or a transcendental number. Therefore, it is not known whether the numbers $\pi$ ${\ displaystyle \ pi}$ and $e$ ${\ displaystyle e}$ algebraically independent . ^[6] ^[32] ^[33] ^[34] ^[35] .
It is not known whether ${^{n}\pi }$ ${\ displaystyle {^ {n} \ pi}}$ integer for any positive integer $n$ ${\ displaystyle n}$ (see tetration ).
It is not known whether ${\frac {1}{\pi }}$ ${\ displaystyle {\ frac {1} {\ pi}}}$ to the ring of periods .
So far, nothing is known about the normality of the number $\pi$ ${\ displaystyle \ pi}$ ; it is not even known which of the digits 0-9 occur in the decimal representation of a number $\pi$ ${\ displaystyle \ pi}$ infinite number of times.

Buffon's Needle Method

A needle is randomly thrown onto a plane lined with equidistant lines, the length of which is equal to the distance between adjacent lines, so that with each throw the needle either does not cross the lines or intersects exactly one. It can be proved that the ratio of the number of intersections of a needle with a line to the total number of throws tends to ${\frac {2}{\pi }}$ ${\ displaystyle {\ frac {2} {\ pi}}}$ with increasing number of throws to infinity ^[36] . This needle method is based on probability theory and underlies the Monte Carlo method ^[37] .

Mnemonic Rules

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

So that we don’t make mistakes,
It is necessary to read correctly:
Three fourteen fifteen
Ninety two and six.

You just have to try
And remember everything as it is:
Three fourteen fifteen
Ninety two and six.

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

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

Memorization can help observance of the poetic size:

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

I know this and remember perfectly:
Pi many signs are superfluous to me, in vain.
We trust in enormous knowledge
Those pi who counted the numbers of the armada.

Learn and Know Among the Famous
Behind the number, how to note luck.

Once at Kolya and Arina
We scattered feather-beds.
White fluff flew around
I swam, froze,
Satisfied
He gave us
Headache of old women.
Wow, dangerous fluff spirit!
- George Alexandrov

Similar verses existed in pre-reform spelling . For example, the following poem composed by the teacher of the Nizhny Novgorod gymnasium Shenrok ^[38] :

Who is joking and will wish soon
Pi find out, the number already knows.

Additional Facts

World record for memorizing number signs $\pi$ ${\ displaystyle \ pi}$ after the decimal point, it belongs to the 21-year-old Indian student Rajveer Meena, who in March 2015 reproduced 70,000 decimal places in 9 hours 27 minutes. ^[39] Prior to this, for almost 10 years, the record was held by the Chinese Liu Chao, who in 2006 reproduced 67,890 decimal places without error within 24 hours and 4 minutes ^[40] ^[41] . In the same 2006, the Japanese Akira Haraguchi said that he remembered the number $\pi$ ${\ displaystyle \ pi}$ up to the 100 thousandth decimal place ^[42] , however, this was not officially verified ^[43] . In Russia, the memorization record belongs to Vladimir Kondryakov (13,183 characters) ^[44] .
In the state of Indiana (USA), in 1897, the Bill on pi was issued, legislatively establishing its value equal to 3.2 ^[45] . This bill did not become law due to the timely intervention of a professor at Purdue University who was present at the state legislature during the consideration of this law.
“The number of Pi for Greenland whales is three” is written in the Whaling Handbook of the 1960s ^[46] .
The Super Pi program, which fixes the time for which a given number of characters (up to 32 million) Pi is calculated, can be used to test computer performance .
On March 14, 2019, when the unofficial holiday of pi was celebrated, Google introduced this number with 31.4 trillion decimal places. Emma Haruka-Iwao, a Google employee in Japan, was able to calculate it with such accuracy. ^[47]

In Culture

There is a feature film named after the number Pi.
The unofficial holiday " Pi Day " is annually celebrated on March 14 , which in the American date format (month / day) is recorded as 3.14, which corresponds to the approximate value of the number $\pi$ ${\ displaystyle \ pi}$ . It is believed ^[48] that the holiday was invented in 1987 by the physicist from San Francisco Larry Shaw , who drew attention to the fact 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 related to the number $\pi$ ${\ displaystyle \ pi}$ , является 22 июля , которое называется «Днём приближённого числа Пи» ( англ. Pi Approximation Day ), так как в европейском формате дат этот день записывается как 22/7, а значение этой дроби является приближённым значением числа $\pi$ $\pi$ .

Notes

↑ PI
↑ Это определение пригодно только для евклидовой геометрии . В других геометриях отношение длины окружности к длине её диаметра может быть произвольным. Например, в геометрии Лобачевского это отношение меньше, чем $\pi$ $\pi$ .
↑ Lambert, Johann Heinrich . Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, С. 265–322.
↑ Доказательство Клейна приложено к работе «Вопросы элементарной и высшей математики», ч. 1, вышедшей в Гёттингене в 1908 году.
↑ Weisstein, Eric W. Постоянная Гельфонда (англ.) на сайте Wolfram MathWorld .
↑ ¹ ² Weisstein, Eric W. Иррациональное число (англ.) на сайте Wolfram MathWorld .
↑ Модулярные функции и вопросы трансцендентности
↑ Ромер П. Новое выражение для π (рус.) // В.О.Ф.Э.М. . — 1890. — № 97 . — С. 2—4 .
↑ Weisstein, Eric W. Pi Squared (англ.) на сайте Wolfram MathWorld .
↑ Гнездовский Ю. Ю. Введение // Справочник по тригонометрии. — Экоперспектива, 2006. — С. 3. — ISBN 985-469-141-1 .
↑ Кымпан Ф., История числа Пи, М., Наука, 1971.
↑ EM Bruins, Quelques textes mathématiques de la Mission de Suse , 1950.
↑ Стройк. Д. Я. Краткий очерк истории математики. М., Наука, 1984, с. 47-48.
↑ Жуков А. В. О числе Пи. М.: МЦНМО, 2002.
↑ Кымпан Ф., История числа Пи, М., Наука, 1971. с. 81.
↑ Исаак Ньютон. Математические работы (в переводе и переработке Мордухай-Болтовского) / Мордухай-Болтовской (также перевод и комментарии). — Москва, Ленинград: Главное изд-во технико-теоретической литературы, 1937.
↑ В наши дни с помощью ЭВМ число $\pi$ $\pi$ вычислено с точностью до триллионов знаков, что представляет скорее технический, чем научный интерес, потому что такая точность практической пользы не представляет.
Точность вычисления ограничивается обычно наличными ресурсами компьютера, — чаще всего временем, несколько реже — объёмом памяти.
↑ Brent, Richard (1975), Traub, JF, ed., " Multiple-precision zero-finding methods and the complexity of elementary function evaluation ", Analytic Computational Complexity (New York: Academic Press): 151–176 , < http://wwwmaths.anu.edu.au/~brent/pub/pub028.html > (eng.)
↑ Jonathan M Borwein. Pi: A Source Book. — Springer, 2004. — ISBN 0387205713 . (eng.)
↑ ¹ ² David H. Bailey, Peter B. Borwein, Simon Plouffe. On the Rapid Computation of Various Polylogarithmic Constants // Mathematics of Computation. — 1997. — Т. 66 , вып. 218 . — С. 903—913 . (eng.)
↑ Fabrice Bellard . A new formula to compute the n ^th binary digit of pi (англ.) . Дата обращения 11 января 2010. Архивировано 21 августа 2011 года.
↑ Simon Plouffe. Indentities inspired by Ramanujan's Notebooks (part 2) (англ.) . Дата обращения 11 января 2010. Архивировано 21 августа 2011 года.
↑ Установлен новый рекорд точности вычисления числа π
↑ Pi Computation Record
↑ Число «Пи» рассчитано с рекордной точностью
↑ 5 Trillion Digits of Pi — New World Record (англ.)
↑ Определено 10 триллионов цифр десятичного разложения для π
↑ Round 2… 10 Trillion Digits of Pi
↑ Хоакин Наварро, 2014 , с. 11..
↑ Weisstein, Eric W. Мера иррациональности (англ.) на сайте Wolfram MathWorld .
↑ Max A. Alekseyev On convergence of the Flint Hills series , 2011.
↑ Weisstein, Eric W. Pi (англ.) на сайте Wolfram MathWorld .
↑ Some unsolved problems in number theory
↑ Weisstein, Eric W. Трансцендентное число (англ.) на сайте Wolfram MathWorld .
↑ An introduction to irrationality and transcendence methods
↑ Обман или заблуждение? Квант № 5 1983 год
↑ Г. А. Гальперин. Биллиардная динамическая система для числа пи .
↑ «Элементарная геометрия» Киселёва стр. 225
↑ 21-Year-Old Memorises 70,000 Pi Digits, Sets Guinness Record
↑ 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, 17.12.2006.
↑ Pi World Ranking List
↑ Историк рассказал, как запомнил 13 тысяч знаков в числе «Пи» — МК.RU, 15 марта 2016
↑ The Indiana Pi Bill, 1897 (англ.)
↑ В. И. Арнольд любит приводить этот факт, см. например книгу Что такое математика ( ps ), стр. 9.
↑ Значение числа «пи» вычислили до 31,4 трлн знаков после запятой (рус.) . www.mk.ru. Date of treatment March 14, 2019.
↑ Статья в Los Angeles Times «Желаете кусочек $\pi$ $\pi$ »? (название обыгрывает сходство в написании числа $\pi$ $\pi$ и слова pie (англ. пирог)) Архивная копия от 19 февраля 2009 на Wayback Machine (недоступная ссылка с 22-05-2013 [2277 дней] — история , копия ) (англ.) .

Literature

Жуков А. В. О числе π . — М. : МЦМНО, 2002. — 32 с. — ISBN 5-94057-030-5 .
Жуков А. В. Вездесущее число «пи». - 2nd ed. — М. : Издательство ЛКИ, 2007. — 216 с. — ISBN 978-5-382-00174-6 .
Наварро, Хоакин. Секреты числа $\pi .$ $\pi .$ Почему неразрешима задача о квадратуре круга. — М. : Де Агостини, 2014. — 143 с. — (Мир математики: в 45 томах, том 7). — ISBN 978-5-9774-0629-1 .
Перельман Я. И. Квадратура круга. — Л. : Дом занимательной науки, 1941. . Переиздание: ЁЁ Медиа, ISBN 978-5-458-62773-3 .
Флорика Кымпан. История числа пи. — М. : Наука, 1971. — 217 с. - 70,000 copies.
David H. Bailey, Jonathan M. Borwein. Pi: The Next Generation A Sourcebook on the Recent History of Pi and Its Computation. — Springer, 2016. — 507 с. — ISBN 978-3-319-32375-6 .

Links

Weisstein, by Eric W. Pi Formulas, (Eng.) In the Wolfram site MathWorld .
Different representations of pi in Wolfram Alpha
sequence A000796 in OEIS
10 million digits of pi
22.4 trillion pi digits (world record)

[1] PI

[2] Это определение пригодно только для евклидовой геометрии . В других геометриях отношение длины окружности к длине её диаметра может быть произвольным. Например, в геометрии Лобачевского это отношение меньше, чем $\pi$ $\pi$ .

[3] Lambert, Johann Heinrich . Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, С. 265–322.

[4] Доказательство Клейна приложено к работе «Вопросы элементарной и высшей математики», ч. 1, вышедшей в Гёттингене в 1908 году.

[5] Weisstein, Eric W. Постоянная Гельфонда (англ.) на сайте Wolfram MathWorld .

[autogenerated1-6] ¹ ² Weisstein, Eric W. Иррациональное число (англ.) на сайте Wolfram MathWorld .

[7] Модулярные функции и вопросы трансцендентности

[8] Ромер П. Новое выражение для π (рус.) // В.О.Ф.Э.М. . — 1890. — № 97 . — С. 2—4 .

[9] Weisstein, Eric W. Pi Squared (англ.) на сайте Wolfram MathWorld .

[10] Гнездовский Ю. Ю. Введение // Справочник по тригонометрии. — Экоперспектива, 2006. — С. 3. — ISBN 985-469-141-1 .

[11] Кымпан Ф., История числа Пи, М., Наука, 1971.

[12] EM Bruins, Quelques textes mathématiques de la Mission de Suse , 1950.

[13] Стройк. Д. Я. Краткий очерк истории математики. М., Наука, 1984, с. 47-48.

[14] Жуков А. В. О числе Пи. М.: МЦНМО, 2002.

[15] Кымпан Ф., История числа Пи, М., Наука, 1971. с. 81.

[16] Исаак Ньютон. Математические работы (в переводе и переработке Мордухай-Болтовского) / Мордухай-Болтовской (также перевод и комментарии). — Москва, Ленинград: Главное изд-во технико-теоретической литературы, 1937.

[17] В наши дни с помощью ЭВМ число $\pi$ $\pi$ вычислено с точностью до триллионов знаков, что представляет скорее технический, чем научный интерес, потому что такая точность практической пользы не представляет.
Точность вычисления ограничивается обычно наличными ресурсами компьютера, — чаще всего временем, несколько реже — объёмом памяти.

[18] Brent, Richard (1975), Traub, JF, ed., " Multiple-precision zero-finding methods and the complexity of elementary function evaluation ", Analytic Computational Complexity (New York: Academic Press): 151–176 , < http://wwwmaths.anu.edu.au/~brent/pub/pub028.html > (eng.)

[19] Jonathan M Borwein. Pi: A Source Book. — Springer, 2004. — ISBN 0387205713 . (eng.)

[bbpf-20] ¹ ² David H. Bailey, Peter B. Borwein, Simon Plouffe. On the Rapid Computation of Various Polylogarithmic Constants // Mathematics of Computation. — 1997. — Т. 66 , вып. 218 . — С. 903—913 . (eng.)

[21] Fabrice Bellard . A new formula to compute the n ^th binary digit of pi (англ.) . Дата обращения 11 января 2010. Архивировано 21 августа 2011 года.

[22] Simon Plouffe. Indentities inspired by Ramanujan's Notebooks (part 2) (англ.) . Дата обращения 11 января 2010. Архивировано 21 августа 2011 года.

[23] Установлен новый рекорд точности вычисления числа π

[24] Pi Computation Record

[25] Число «Пи» рассчитано с рекордной точностью

[Рекорд-26] 5 Trillion Digits of Pi — New World Record (англ.)

[27] Определено 10 триллионов цифр десятичного разложения для π

[pi-10t-28] Round 2… 10 Trillion Digits of Pi

[_277074e4cc935ce7-29] Хоакин Наварро, 2014 , с. 11..

[30] Weisstein, Eric W. Мера иррациональности (англ.) на сайте Wolfram MathWorld .

[31] Max A. Alekseyev On convergence of the Flint Hills series , 2011.

[32] Weisstein, Eric W. Pi (англ.) на сайте Wolfram MathWorld .

[33] Some unsolved problems in number theory

[34] Weisstein, Eric W. Трансцендентное число (англ.) на сайте Wolfram MathWorld .

[35] An introduction to irrationality and transcendence methods

[36] Обман или заблуждение? Квант № 5 1983 год

[37] Г. А. Гальперин. Биллиардная динамическая система для числа пи .

[38] «Элементарная геометрия» Киселёва стр. 225

[39] 21-Year-Old Memorises 70,000 Pi Digits, Sets Guinness Record

[40] Chinese student breaks Guiness record by reciting 67,890 digits of pi

[41] Interview with Mr. Chao Lu

[42] How can anyone remember 100,000 numbers? — The Japan Times, 17.12.2006.

[43] Pi World Ranking List

[44] Историк рассказал, как запомнил 13 тысяч знаков в числе «Пи» — МК.RU, 15 марта 2016

[45] The Indiana Pi Bill, 1897 (англ.)

[46] В. И. Арнольд любит приводить этот факт, см. например книгу Что такое математика ( ps ), стр. 9.

[47] Значение числа «пи» вычислили до 31,4 трлн знаков после запятой (рус.) . www.mk.ru. Date of treatment March 14, 2019.

[48] Статья в Los Angeles Times «Желаете кусочек $\pi$ $\pi$ »? (название обыгрывает сходство в написании числа $\pi$ $\pi$ и слова pie (англ. пирог)) Архивная копия от 19 февраля 2009 на Wayback Machine (недоступная ссылка с 22-05-2013 [2277 дней] — история , копия ) (англ.) .