Pi

The mathematical constant ? is the ratio of a circle's circumference (Greek ??????????, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. The name of the Greek letter ? is pi (pronounced pie), and this spelling can be used in typographical contexts where the Greek letter is not available. ? is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number.

Numerical approximations of π

Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π.

Related Topics:
Approximation - Significant figures - Continued fraction

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An Egyptian scribe named Ahmes wrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes stated that he copied a Middle Kingdom papyrus—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160.

Related Topics:
Ahmes - Rhind Mathematical Papyrus - Egyptian - Second Intermediate Period - Middle Kingdom - Papyrus

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The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.

Related Topics:
Liu Hui - 263

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The Indian mathematician and astronomer Aryabhata gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.

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The Chinese mathematician and astronomer Zu Chongzhi computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the 5th century.

Related Topics:
Zu Chongzhi - 5th century

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The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

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:2 π = 6.2831853071795865

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The German mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tombstone.

Related Topics:
Ludolph van Ceulen - 1600 - Tombstone

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The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today.

Related Topics:
Jurij Vega - 1789 - 1841 - William Rutherford - John Machin - 1706

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None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as Machin's:

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: rac{pi}{4} = 4 rctanrac{1}{5} - rctanrac{1}{239}

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together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with

Related Topics:
Taylor series - Arctan - Polar coordinates - Complex number

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:(5+i)^4cdot(-239+i)=-114244-114244i.

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Formulas of this kind are known as Machin-like formulas.

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Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used in the past.

Related Topics:
Gauss-Legendre algorithm - Borwein's algorithm - Salamin-Brent algorithm - 1976

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The first one million digits of π and 1/π are available from Project Gutenberg (see external links below).

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The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulas were used for this:

Related Topics:
2002 - Yasumasa Kanada - Tokyo University - Hitachi - Supercomputer

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: rac{pi}{4} = 12 rctanrac{1}{49} + 32 rctanrac{1}{57} - 5 rctanrac{1}{239} + 12 rctanrac{1}{110443}

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:K. Takano (1982).

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: rac{pi}{4} = 44 rctanrac{1}{57} + 7 rctanrac{1}{239} - 12 rctanrac{1}{682} + 24 rctanrac{1}{12943}

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:F. C. W. Störmer (1896).

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These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records.

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In 1996 David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series:

Related Topics:
1996 - David H. Bailey - Peter Borwein - Simon Plouffe - Infinite series

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: pi = sum_{k = 0}^{infty} rac{1}{16^k}

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left( rac{4}{8k + 1} - rac{2}{8k + 4} - rac{1}{8k + 5} - rac{1}{8k + 6} ight)

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This formula permits one to easily compute the kth binary or hexadecimal digit of π, without

Related Topics:
Binary - Hexadecimal

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having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0).

Related Topics:
Programming languages - PiHex - Quadrillion

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Other formulas that have been used to compute estimates of π include:

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:

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rac{pi}{2}=

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sum_{k=0}^inftyrac{k!}{(2k+1)!!}=

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1+rac{1}{3}left(1+rac{2}{5}left(1+rac{3}{7}left(1+rac{4}{9}(1+...) ight) ight) ight)

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:Newton.

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: rac{1}{pi} = rac{2sqrt{2}}{9801} sum^infty_{k=0} rac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}

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:Ramanujan.

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This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.

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: rac{1}{pi} = 12 sum^infty_{k=0} rac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}

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:David Chudnovsky and Gregory Chudnovsky.

Related Topics:
David Chudnovsky - Gregory Chudnovsky

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: {pi} = 20 rctanrac{1}{7} + 8 rctanrac{3}{79}

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:Euler.

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On computers running Microsoft Windows OS, the program PiFast can be used to quickly calculate a large amount of digits. The largest number of digits of π calculated on a home computer, 25,000,000,000, was calculated with PiFast in 17 days.

Related Topics:
Microsoft Windows - OS - PiFast

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Miscellaneous formulas

In base 60, π can be approximated to eight significant figures as

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: 3 + rac{8}{60} + rac{29}{60^2} + rac{44}{60^3}

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In addition, the following expressions can be used to estimate π

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• accurate to 9 digits:

:(63/25)((17+15sqrt 5)/(7+15sqrt5))

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• accurate to 17 digits:

:3 + rac{48178703}{340262731}

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• accurate to 3 digits:

:sqrt{2} + sqrt{3}

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:Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of right triangles which are either isosceles or halves of equilateral triangles.

Related Topics:
Karl Popper - Plato - Right triangle - Isosceles - Equilateral

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Less accurate approximations

In 1897, a physician and amateur mathematician from Indiana named Edward J. Goodwin believed that the transcendental value of π was wrong. He proposed a bill to Indiana Representative T. I. Record which expressed the "new mathematical truth" in several ways:

Related Topics:
Indiana - Edward J. Goodwin - Transcendental - T. I. Record

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:The ratio of the diameter of a circle to its circumference is 5/4 to 4. (π = 3.2)

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:The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7. (π ≈ 3.23...)

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:The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. (π = 4)

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:It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. (π ≈ 9.24 if rectangle is emended to triangle; if not, as above.)

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The bill also recites Goodwin's previous accomplishments: "his solutions of the trisection of the angle, doubling the cube having been already accepted as contributions to science by the American Mathematical Monthly....And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend." These false claims are typical of a mathematical crank. The claims trisection of an angle and the doubling of the cube are particularly widespread in crank literature.

Related Topics:
Trisection of the angle - Doubling the cube - American Mathematical Monthly - Crank

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The Indiana Assembly referred the bill to the Committee on Swamp Lands, which Petr Beckmann has seen as symbolic. It was transferred to the Committee on Education, which reported favorably, and the bill passed unanimously. One argument used was that Goodwin had copyrighted his discovery, and proposed to let the State use it in the public schools for free. As this debate concluded, Professor C. A. Waldo arrived in Indianapolis to secure the annual appropriation for the Indiana Academy of Sciences. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already knew as many crazy people as he cared to.

Related Topics:
Assembly - Petr Beckmann - C. A. Waldo - Indianapolis - Indiana Academy of Sciences

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The Indiana Senate had not yet finally passed the bill (which they had referred to the Committee on Temperance), and Professor Waldo coached enough Senators overnight that they postponed the bill indefinitely. source

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