Pi

Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
Area of a circle Circumference Use in other formulae
Properties
Irrationality Transcendence
Value
Less than 22/7 Approximations Madhava's correction term Memorization
People
Archimedes Liu Hui Zu Chongzhi Aryabhata Madhava Jamshīd al-Kāshī Ludolph van Ceulen François Viète Seki Takakazu Takebe Kenko William Jones John Machin William Shanks Srinivasa Ramanujan John Wrench Chudnovsky brothers Yasumasa Kanada
History
Chronology A History of Pi
In culture
Indiana pi bill Pi Day
Related topics
Squaring the circle Basel problem Six nines in π Other topics related to π
v t e

π (sometimes written pi) is a mathematical constant that is the ratio of a circle's circumference to its diameter. π is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed as a ratio of two integers (such as 22/7); consequently its decimal representation never ends and never repeats. π is a transcendental number: an irrational number that also cannot be produced with a finite sequence of algebraic operations (powers, roots, sums, etc.). The transcendence of π means that it is impossible to solve the ancient challenge of squaring the circle. The digits in any representation of π appear to be "random", although no proof of this supposed randomness has yet been discovered.

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially concerning circles, ellipses, or spheres. π is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants.

For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Mathematicians associated with this effort include Archimedes, Leonhard Euler, Carl Friedrich Gauss, Isaac Newton, Ramanujan, and John von Neumann. In the 20th century, mathematicians and computer scientists discovered new algorithms which – when combined with increasing computer speeds – produced a steady stream of world records extending the decimal representation of π, leading to over a trillion (10¹²) digits in 2012. Scientific applications require no more than a few hundred digits of π, so the primary motivation is the human desire to break records; but the extensive calculations involved are also used to test supercomputers and high-precision multiplication algorithms.

The peculiar properties of π, coupled with its widespread use in the science and engineering, have contributed to its popularity outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports on new records in its precise calculation. In the past century, several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.

The decimal representation of π truncated to 100 decimal places is π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 …^[1] The hexadecimal approximation to 20 digits is ^[2] A sexagesimal (base 60) approximation is 3:8:30.^[3] Fractions that can be used to approximate π include (in order of increasing accuracy) ⁠22/7⁠, ⁠333/106⁠, ⁠355/113⁠, ⁠52163/16604⁠, and ⁠103993/33102⁠.^[4]

π is commonly defined as the ratio of a circle's circumference C to its diameter d:^[5] ${\displaystyle \pi ={\frac {C}{d}}.}$

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d. This definition is only valid in Euclidean plane geometry, because it relies on the assumption that all circles are similar. Another definition of π, which also relies on Euclidean geometry, involves the area and radius of a circle: ${\displaystyle \scriptstyle \pi ={\mathit {Area}}/{\mathit {radius}}^{2}.}$^[5]

π is sometimes defined using trigonometric functions to avoid dependence on Euclidean geometry. Such definitions include:^[5][6]

• π is the smallest positive x for which cosine(x) equals −1 (π = arccos(−1))
• π is twice the smallest positive x for which sine(x) equals 1 (π = 2 arcsin(1))
• π is four times the smallest positive x for which tangent(x) equals 1 (π = 4 arctan(1))

Mathematicians use the Greek letter π to represent the ratio of a circle's circumference to its diameter. π is always written in lower-case, even at the beginning of a sentence, because the capital Greek letter Template:PI has a completely different meaning in mathematics as the product of a sequence. In some fonts though, a lower-case π lacks curves and looks very similar to a capital Template:PI. The Greek letter π can be represented by the Latin word pi, which is also used to represent the same ratio.^[7] In English, π is pronounced as "pie" ( /paɪ/, /ˈpaɪ/).^[8]

The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in 1706 in his work Synopsis Palmariorum Matheseeos; or, a New Introduction to the Mathematics.^[9] Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius 1. He may have chosen π because it was the first letter in the Greek spelling of the word periphery.^[10] Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have used the Greek letter before Jones.^[11] The Greek letter had been used earlier for geometric concepts, for example, in 1631 it was used by William Oughtred to represent the half circumference of a circle.^[11]

After Jones introduced the Greek letter in 1706, it was not used by other mathematicians until Euler used it in 1736. Prior to 1736, mathematicians typically used letters such as c or p to represent the ratio of the circumference to diameter.^[11] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.^[11] In 1748, Euler used π in his widely-read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and thereafter the Greek letter was universally used in the Western world.^[11]

π is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7.^[12] The exact irrationality measure of π is not known, however it has been estimated as 7.6063.^[13]

π is a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root.^[14] One consequence of the transcendence of π is that it cannot be expressed using any combination of rational numbers and square roots or n-th roots such as ${\displaystyle \scriptstyle {\sqrt[{3}]{31}}}$.

A consequence of the transcendence of π is the fact that it is not possible to "square the circle", meaning it is not possible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This follows from the transcendental property, because no transcendental numbers can be constructed with the compass and straightedge technique.^[15] Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.^[16]

The digits of π appear to be random, with no observable pattern. A mathematical test for randomness is normality, meaning that all possible sequences of digits (of any given length) are equally likely.^[17] The hypothesis that π is normal has not been proven or disproven.^[18][19] Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.^[20] Despite the fact that π's digits are statistically random, it contains some sequences of digits that appear non-random to the layman, such as the Feynman point, which is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of π.^[21]

It is unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of π, e^π, and Γ(1/4) in 1996.^[22]

For most calculations involving π, a handful of digits provide sufficient precision. Thirty-nine digits are sufficient to support most cosmological calculations, because that is the accuracy which is necessary to calculate the diameter of the universe with a precision of one atom.^[23] Accounting for additional digits needed to compensate for computational round-off errors, a few hundred digits would suffice for any scientific application.^[23] Despite this, mathematicians have worked strenuously to compute π to thousands and millions of digits.^[24] The desire for large number of digits may be partly ascribed to the human compulsion to break records, as new records for computing π often make news headlines around the world.^[25][26] Computing a large number of digits of π does have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms), and providing raw data to evaluate the randomness or normality of the digits of π.^[27]

The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 is approximately equal to 2π. A few pyrimdologists conclude from this value that the pyramid builders had knowledge of π and deliberately designed the pyramid to incorporate the value.^[28] However, mainstream historians believe that ancient Egyptians had no concept of π and that it is merely a coincidence that the ratio of perimeter to height is about 2π.^[29]

The earliest written approximations of π are found in Babylon and Egypt, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8.^[30] In Egypt, the Rhind Papyrus, dated around 1650 BC, has a formula for the area of circle that treats π as (16/9)².^[30]

In India, around 600 BC, the ancient Indian math texts Shulba Sutras, written in Sanskrit, treat π as (9785/5568)².^[31] In 150 BC, or perhaps earlier, Indian sources treat π as ${\displaystyle \scriptstyle {\sqrt {10}}.}$^[32]

The Hebrew Bible, written between 8th and 3rd centuries BC, contains two verses which suggest that π had a value of three. The two verses, 1 Kings 7:23 and 2 Chronicles 4:2, discuss a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits.^[33] Suggestions that the pool had a hexagonal shape or an outward curving rim have been offered to explain the disparity.^[34]

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons which was used around 250 BC by Greek mathematician Archimedes.^[35] This polygonal algorithm remained the primary approach for computing π for over 1,000 years.^[36] Archimedes computed upper and lower bounds of π by drawing regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons.^[37] By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 22/7.^[37] Archimedes' upper bound of 22/7 may have led to widespread belief that π was equal to 22/7.^[38] Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.^[39] Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.^[40]

In ancient China, values for π included 3.1547 (around 0 AD), ${\displaystyle \scriptstyle {\sqrt {10}}}$ (100 AD), and 142/45 (third century).^[41] Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain an value of π of 3.1416.^[42][43] Hui later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.^[42] The Chinese mathematician Zu Chongzhi, around 480, calculated that π ≈ 355/113 using Liu Hui's algorithm applied to a 12,288-sided polygon. This value would remain the most accurate approximation of π available for the next 800 years.^[44]

In India, in 499, astonomer Aryabhata in his work Aryabhatiya had value 3.1416.^[45] Fibonacci in circa 1220 computed 3.1418 using a polygonal method, independent from Archimedes.^[46] Italian author Dante apparently employed the value ${\displaystyle \scriptstyle 3+{\sqrt {2}}/10}$.^[46]

Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1430 using a polygon with 3 x 2²⁸ sides, which stood as the world record for about 180 years.^[47] French mathematician Francois Viete in 1579 achieved 9 digits used polygon of 3 x 2¹⁷ sides.^[47] Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. ^[47] In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was was called the "Ludolphian number" in Germany until the early 20th century).^[48] Dutch scientist Willebrord Snellius reached 34 digits in 1621,^[49] and Austrian astronomer Christoph Grienberger arrived at 39 digits in 1630, which would remain the most accurate approximation manually achieved using polygonal algorithms.^[49]

The calculation of π was revolutionized in the 16th and 17th centuries by the discovery of infinite series. These series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for π most notably by European mathematicians such as James Greory and Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD.^[50]

The first written description of an infinite series which could be used to compute π was written in Sanskrit verse by Indian astronomer Nilakantha Somayaji in the work Tantrasamgraha, dating from around 1500 AD.^[51] The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an eariler Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425.^[51] Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Leibniz formula for π.^[51] Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician, Jamshīd al-Kāshī using a polygonal algorithm.^[52] The Leibniz formula for π is simple, but converges very slowly, so is not used in modern π calculations.^[53]

The first infinite series discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593.^[54] The second infinite series found in Europe, by John Wallis in 1655, also was an infinite product.^[54] The discovery of calculus by English scientist Isaac Newton and German mathematician Leibniz in the 1660s, created the foundation for a large number of infinite series which could be exploited by π hunters. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time".^[55]

Infinite series based on the arctan function generally converge much faster than other infinite series, and were used to set records for the next 300 years, culminating in a 620 digit approximation in 1946 by Daniel Ferguson – the best approximation calculated without the aid of a calculating device.^[56] Scottish mathematician James Gregory produced the first arctan series in Europe in 1671, and Leibniz obtained the same series in 1674.^[57] In 1699, English mathematician Abraham Sharp used Leibniz's series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.^[58] In 1706 John Machin produced a new arctan-based series that allowed him to reach 100 digits.^[59] Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Gauss.^[60] British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.^[60]

Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. Swiss scientiest Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers, and it cannot be written as a fraction.^[12] French mathematician Adrien-Marie Legendre proved in 1794 that π² is also irrational.

When Euler solved the famous Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a deep connection between π and the prime numbers. This knowledge later contributed to the development and study of the Riemann zeta function.^[61]

Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by German mathematician Ferdinand von Lindemann.^[62]

The development of the computers in the mid twentieth century again revolutionized the hunt for digits of π. American mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator.^[63] Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann used the ENIAC computer to compute 2,037 digits of π in 1949, a calculation that took 70 hours of computer time.^[64] The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.^[65]

Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing pi, which were much faster than the infinite series; and second, new multiplication algorithms which could multiply large numbers together very rapidly.^[66] The fast multiplication algorithms are particularly important in computer-based π computations, because the majority of the computer's time is typically spent performing multiplications.^[67] Fast multiplication algorithms include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform based methods.^[68]

The iterative algorithms were independently published in 1975–1976 by Australian scientist Richard Brent and Eugene Salamin.^[69] These algorithms were unique because they utilized an iterative approach rather than an infinite series. However, Salamin and Brent were not the first to discover the approach: it was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss–Legendre algorithm.^[69] The algorithm, as modified by Salamin and Brent, is also referred to as the "Brent-Salamin algorithm".

The iterative algorithms were widely used by π hunters following 1980 because they have the potential to be faster than infinite series algorithms: Whereas infinite series typically increase the number of digits by a fixed amount for each added term, iterative algorithms typically multiply the number of digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers (Jonathan Borwein and Peter Borwein) produced an iterative algorithm that quadruples the number of digits in each step; and in 1987 they discovered an iterative algorithm that increases the number of digits five times each iteration.^[70] Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing π between 1995 and 2002.^[71] The rapid convergence of iterative algorithms comes at a price: the iterative algorithms require significantly more memory usage than infinite series.^[71]

Modern π calculators do not use iterative AGM-based algorithms exclusively. New infinite series were discovered in the 1980s and 1990s which are as fast as iterative algorithms, yet are simpler and use less memory.^[71] The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulas for π, remarkable for their elegance, mathematical depth and rapid convergence.^[72] One of the formulae, based on modular equations, was:

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

This equation converges much more rapidly than most arctan series, including Machin's formula.^[73] Ramanujan's formula was not used by π hunters until Bill Gosper used it in 1985 to set a record of 17 million digits.^[74] Ramanujan's formulas anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.^[75] The Chudnovsky algorithm developed by the Chudnovsky brothers in 1987 is:

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

which produces 14 digits of π per term.^[76] The Chudnovsky algorithm has been used for several record-setting π calculations including the first calculation of over one billion (10⁹) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×10¹²) digits by Fabrice Bellard in 2009, and 10 trillion (10¹³) digits in 2011 by Alexander Yee and Shigeru Kondo.^[77][78]

In 2006, Canadian mathematician Simon Plouffe, using the integer relation algorithm PSLQ, found a series of formulas for π represented by the following general equation:

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

Where ${\displaystyle {\mathit {q}}}$ is e^π (Gelfond's constant), and where ${\displaystyle {\mathit {k}}}$ is an odd number, and ${\displaystyle {\mathit {a,b,c}}}$ are rational numbers.^[79]

Two algorithms were discovered in 1995 which opened up new avenues of research into π. The algorithms are characterized as spigot algorithms because they can produce a sequence of single digits of π, as drops of water from a spigot.^[80] This is in contrast to infinite series or iterative algorithms, which require the operator to predetermine a fixed number of digits to compute, and no digits are produced until the algorithm has finished and generated all the digits.^[80]

American mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995 which generated digits of π in order: 3,1,4,1,5, ....^[81] When the algorithm is executed, it begins generating digits immediately, providing instant feedback to the operator.^[81] The algorithm's speed is comparable to arctan algorithms but not as fast as iterative algorithms.^[81]

Another spigot algorithm from 1995 is the the BBP digit extraction algorithm discovered by Simon Plouffe.^[82][83] The formula was a breakthrough for π hunters because it can produce any individual hexadecimal digit of π without calculating all the preceding digits.^[82] From the hexadecimal digit, octal or binary digits may be readily extracted. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found which rapidly produces decimal digits.^[84] An important application of digit extraction algorithms is to validate new claims of record π computations: after a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several random hexadecimal digits near the end, and if they match, it provides a measure of confidence that the entire computation is correct.^[78]

Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (10¹⁵th) bit of π, which turned out to be 0.^[85] In September 2010, a Yahoo! employee used the company's Hadoop application on 1,000 computers over a 23-day period to compute 256 bits of π at the two-quadrillion^th (2×10¹⁵th) bit.^[86]

For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr². π appears in formulas for areas and volumes of most other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori.^[87]

• Circumference of a circle with radius ${\displaystyle r}$ ${\displaystyle =2\pi r}$
• Area of a circle with radius ${\displaystyle r}$ ${\displaystyle =\pi r^{2}}$
• Volume of a sphere with radius ${\displaystyle r}$ ${\displaystyle ={\tfrac {4}{3}}\pi r^{3}}$
• Surface area of a sphere with radius ${\displaystyle r}$ ${\displaystyle =4\pi r^{2}}$

π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. For example, an integral which specifies half the area of the unit disk is given by:^[88]

${\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}}$

and an integral which evaluates half the circumference of the unit circle is:^[87]

${\displaystyle \int _{-1}^{1}{\frac {1}{\sqrt {1-x^{2}}}}\,dx=\pi }$

The trigonometric functions sine and cosine have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π radians, so 1° = (π/180) radians.^[89]

A complex number ${\displaystyle z}$ can be expressed in polar coordinates (r, φ) as follows:^[90]

${\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi )}$

The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula:^[91]

${\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi \!}$

where i is the imaginary unit satisfying i² = −1 and e is the base of the natural logarithm. This formula implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these rotations have a period of 2π radians. In particular, the half rotation φ = π radians results in Euler's identity, celebrated by mathematicians because it contains several important mathematical constants:^[91]

${\displaystyle e^{i\pi }+1=0.\!}$

There are n different n-th roots of unity^[92]

${\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}$

The Gaussian integral is:^[93]

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

Related to the Gaussian integral is the gamma function, which is a function that extends the concept of factorial to all real numbers. π is found in the result when the gamma function is evaluated at half-integers: ${\displaystyle \scriptstyle \Gamma (1/2)={\sqrt {\pi }}}$ and ${\displaystyle \scriptstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}$.^[94] The gamma function can be used to create a simple approximation to ${\displaystyle \scriptstyle n!}$ for large ${\displaystyle \scriptstyle n}$: ${\displaystyle \scriptstyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\!}$ which is known as Stirling's approximation.^[95]

An occurrence of π in the Mandelbrot set fractal was discovered by American David Boll in 1991.^[96] He found that when looking at the pinch points of the Mandelbrot set, the number of iterations needed for the point (−.75, ε) before diverging, multiplied by ε, was equal to π. Based on this initial finding, American mathematician Aaron Klebanoff developed a further test near another pinch point (.25,ε) in the Mandelbrot set and found that the number of iterations times the square root of ε was equal to π.^[97][96]

The Riemann zeta function ζ(s) is a function which is utilized in many areas of mathematics. The function is defined for all complex numbers s ≠ 1 and is equal to

${\displaystyle \zeta (s)={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \!}$

when the real part of s is greater than 1. A factor of π emerges when ζ is evaluated at the positive even integers, for example:

${\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}.\!}$

Finding a closed-form expression for the above series was a famous problem in mathematics called the Basel problem, which was solved by Leonhard Euler in 1735 when he showed it was equal to ${\displaystyle \scriptstyle {\frac {\pi ^{2}}{6}}}$.^[61] The fact that ${\displaystyle \scriptstyle \zeta (2)={\frac {\pi ^{2}}{6}}}$ leads to the number theory result that the probability of two random numbers being relatively prime is equal to ${\displaystyle \scriptstyle 6/\pi ^{2}}$.^[98][99] This result is based on the observation that the probability that any number is divisible by a prime ${\displaystyle \scriptstyle p}$ is ${\displaystyle \scriptstyle 1/p}$ (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is ${\displaystyle \scriptstyle 1/p^{2}}$, and the probability that at least one of them is not is ${\displaystyle \scriptstyle 1-1/p^{2}}$. For distinct primes, these divisibility events are mutually independent, thus the probability that two numbers are relatively prime is given by a product over all primes:^[100]

${\displaystyle \prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.}$

This probability can be used in conjunction with a random number generator to approximate π using a Monte Carlo approach.^[101]

Although not a physical constant, π appears routinely in equations describing fundamental principles of the universe, often because of π's relationship to the circle and to spherical coordinate systems. Important physics formulae that include π are:

• The approximate period T of a simple pendulum of length L, swinging with a small amplitude (g is the gravitational force): ^[102]

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}\!}$

• Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δx) and momentum (Δp) can not both be arbitrarily small at the same time:^[103]

${\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}={\frac {\hbar }{2}}}$

• Einstein's field equation is a key part of his general theory of relativity, and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:^[104][105]

${\displaystyle R_{ik}-{g_{ik}R \over 2}+\Lambda g_{ik}={8\pi G \over c^{4}}T_{ik}}$

• Coulomb's law for the electric force, describing the force between two electric charges (q₁ and q₂) separated by distance r (with ε₀ representing the vacuum permittivity of free space):^[106]

${\displaystyle F={\frac {\left|q_{1}q_{2}\right|}{4\pi \varepsilon _{0}r^{2}}}}$

• The Biot–Savart law is used to compute the resultant magnetic field B at position r generated by a steady current I (for example due to a wire): a continual flow of charges which is constant in time and the charge neither accumulates nor depletes at any point. The law is a physical example of a line integral: evaluated over the path C the electric currents flow. The equation in SI units is:^[107]

${\displaystyle \mathbf {B} ={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}}$

• Kepler's third law, which describes two co-orbiting bodies, relates the orbital period P, the gravitational constant G, the semi-major axis a, and the masses (M and m):^[108]

${\displaystyle \left({\frac {2\pi }{P}}\right)^{2}\,a^{3}=G(M+m)}$

π is found in the probability density function for the normal distribution with mean μ and standard deviation σ, encapsulated in the Gaussian integral:^[109]

${\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}}$

Another probability formula which includes π is the probability density function for the (standard) Cauchy distribution:^[110]

${\displaystyle f(x)={\frac {1}{\pi (1+x^{2})}}.}$

π is present in some structural engineering formulae, such as the buckling formula, derived by Euler, that gives the maximum axial load that a long, slender, ideal column can carry without buckling:^[111] ${\displaystyle F_{K}={\frac {\pi ^{2}EI}{s^{2}}}}$

π is also present in Stokes' law, which approximates the frictional force exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid:^[112] ${\displaystyle F_{R}=6\,\pi \,\eta \,r\,v\!}$

The Fourier transform is used for a wide variety of signal processing and image processing applications:^[113] ${\displaystyle F(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,\mathrm {d} t}$

Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the ratio between the actual length of a river and its straight-line from source to mouth length tends to approach π. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist and so on. However, increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting, creating an ox-bow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.^[114][115]

Monte Carlo methods, which evalutate the results of multiple random trials, can be used to create approximations of π.^[116] Buffon's needle is one such technique: If a needle of length l is dropped n times on a surface containing parallel lines drawn t units apart, and if x of those times it comes to rest crossing a line (x > 0), then one may approximate π based on the counts:^[117][118]

${\displaystyle \pi \approx {\frac {2nl}{xt}}.}$

Another Monte Carlo method for computing π is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal ${\displaystyle \scriptstyle \pi /4}$.^[119]

Monte Carlo methods for approximating π are very slow compared to other methods, and are never used to approximate π when speed or accuracy are desired.^[120]

Like all irrational numbers, π can be represented by an infinite simple continued fraction. The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:^[121]

${\displaystyle \pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\ddots }}}}}}}}}}}}}}}$

Truncating the continued fraction at any point generates a fraction that approximates π. Two fractions produced from π's continued fraction, 22/7 and 355/113, have been used historically to approximate π. The continued fraction can be used to generate the best possible rational approximation (that is, no other approximation with a smaller denominator will be closer to π).^[122] Although the simple continued fraction for π does not exhibit a pattern, mathematicians have discovered several generalized continued fractions which do, such as:^[123]

${\displaystyle \pi =\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\ddots }}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\ddots }}}}}}}}}}=\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\ddots }}}}}}}}}}}$

Many persons have memorized large numbers of digits of π, a practice called piphilology.^[124] One common technique is to memorize a story or poem, in which the word-lengths represent the digits of π: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. When a poem is utilized, it is sometimes referred to as a "piem". An early example of such a poem, originally devised by English scientist James Jeans: "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."^[124][125] Poems for memorizing π have been composed in several languages in addition to English.^[126]

The record for memorizing digits of π, certified by Guinness World Records, is 67,890 digits, recited by in China by Lu Chao in 24 hours and 4 minutes on November 20, 2005.^[127][128] In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.^[129] Record-setting π memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the method of loci.^[130]

A few authors have used the digits of π to establish a new form of constrained writing, where the word-lengths are required to represent the digits of π. The Cadaeic Cadenza contains the first 3835 digits of π in this manner,^[131] and a full-length novel has been published which contains 10,000 words, each representing one digit of π.^[132]

Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, π has been represented in popular culture more than most other mathematical constructs. Palais de la Découverte, a science museum in Paris, contains a circular room known as the "pi room". On its wall is inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on a 1853 calculation by English mathematician William Shanks, which included an error in the 528th digit. The error was detected in 1946 and corrected in 1949.^[133]

π and its digital representation are often used by self-described "math geeks" for inside jokes among mathematically and technologically-minded groups. Many schools around the world observe Pi Day (March 14, from 3.14).^[135] Several college cheers at the Massachusetts Institute of Technology include "3.14159!"^[134] During the 2011 auction for Nortel's portfolio of valuable technology patents, Google made a series of unusually specific bids based on mathematical and scientific constants, including π.^[136]

On November 7, 2005, alternative musician Kate Bush released the album Aerial. The album contains the song "Pi" whose lyrics consist principally of Bush singing the digits of π to music, beginning with "3.14".^[137] In Carl Sagan's novel Contact, π played a key role in the story. The novel suggested that there was a message buried deep within the digits of π placed there by the creator of the universe.^[138]

In 1897, an amateur mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi Bill, which described a method to square the circle, and contains text which assumes various incorrect values of π, including 3.2. The bill is notorious as an attempt to establish scientific truth by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate.^[139]

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• "Pi" at Wolfram Mathworld
• Representations of Pi at Wolfram Alpha
• Pi Search Engine 2 billion searchable digits of π, √2, and e

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