Pi: Difference between revisions

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

In Euclidean plane geometry, π may be defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Advanced textbooks define π analytically using trigonometric functions, for example as the smallest positive x for which sin(x) = 0, or as twice the smallest positive x for which cos(x) = 0. All these definitions are equivalent.

The numerical value of π truncated to 50 decimal places (sequence A000796 in the OEIS) is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

See the links below for more digits.

Although this precision is more than sufficient for use in engineering and science, the exact value of π has decimal places that never end. Much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, in addition to supercomputer calculations that have determined over 1 trillion digits of π, no pattern in the digits has ever been found. Digits of π are available from multiple resources on the Internet, and a regular personal computer can compute billions of digits with available software.

Properties

π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert.

π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle.

Formulae involving π

Geometry

${\displaystyle \pi }$ appears in many formulæ in geometry involving circles and spheres.

Geometrical shape	Formula
Circumference of circle of radius r and diameter d	${\displaystyle C=\pi d=2\pi r\,\!}$
Area of circle of radius r	${\displaystyle A=\pi r^{2}\,\!}$
Area of ellipse with semiaxes a and b	${\displaystyle A=\pi ab\,\!}$
Volume of sphere of radius r and diameter d	${\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {1}{6}}\pi d^{3}\,\!}$
Surface area of sphere of radius r	${\displaystyle A=4\pi r^{2}\,\!}$
Volume of cylinder of height h and radius r	${\displaystyle V=\pi r^{2}h\,\!}$
Surface area of cylinder of height h and radius r	${\displaystyle A=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!}$
Volume of cone of height h and radius r	${\displaystyle V={\frac {1}{3}}\pi r^{2}h\,\!}$
Surface area of cone of height h and radius r	${\displaystyle A=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!}$

(All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.)

Also, the angle measure of 180° (degrees) is equal to π radians.

Analysis

Many formulæ in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.

• François Viète, 1593 (proof):

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

• Leibniz' formula (proof):

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

This commonly cited infinite series is usually written as above, but is more concisely expressed as:

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {\pi }{4}}}$

• Wallis product (proof):

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

${\displaystyle \prod _{n=1}^{\infty }{\frac {(2n)^{2}}{(2n)^{2}-1}}=\prod _{n=1}^{\infty }{\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}={\frac {\pi }{2}}}$

• Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and Bailey web page)

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

• An integral formula from calculus (see also Error function and Normal distribution):

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

• Basel problem, first solved by Euler (see also Riemann zeta function):

${\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}$

${\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}}$

and generally, ${\displaystyle \zeta (2n)}$ is a rational multiple of ${\displaystyle \pi ^{2n}}$ for positive integer n

• Gamma function evaluated at 1/2:

${\displaystyle \Gamma \left({1 \over 2}\right)={\sqrt {\pi }}}$

• Stirling's approximation:

${\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}$

• Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"):

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

• Property of Euler's totient function (see also Farey sequence):

${\displaystyle \sum _{k=1}^{n}\phi (k)\sim 3n^{2}/\pi ^{2}}$

• Area of one quarter of the unit circle:

${\displaystyle \int _{0}^{1}{\sqrt {1-x^{2}}}\,dx={\pi \over 4}}$

• An application of the residue theorem

${\displaystyle \oint {\frac {dz}{z}}=2\pi i,}$

where the path of integration is a closed curve around the origin, traversed in the standard counterclockwise direction.

Continued fractions

π has many continued fractions representations, including:

${\displaystyle {\frac {4}{\pi }}=1+{\frac {1}{3+{\frac {4}{5+{\frac {9}{7+{\frac {16}{9+{\frac {25}{11+{\frac {36}{13+...}}}}}}}}}}}}}$

(Other representations are available at The Wolfram Functions Site.)

Number theory

Some results from number theory:

• The probability that two randomly chosen integers are coprime is 6/π².

• The probability that a randomly chosen integer is square-free is 6/π².

• The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.

• The product of (1-1/p²) over the primes, p, is 6/π².${\displaystyle \prod _{p\in \mathbb {P} }\left(1-{\frac {1}{p^{2}}}\right)={\frac {6}{\pi ^{2}}}}$

Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,..., N}, and then take the limit as N approaches infinity.

The fact (note the order to which the number approaches an integer) that

${\displaystyle e^{\pi {\sqrt {163}}}=262537412640768743.99999999999925007...}$

or equivalently,

${\displaystyle e^{\pi {\sqrt {163}}}=640320^{3}+743.99999999999925007...}$

can be explained by the theory of complex multiplication.

Dynamical systems and ergodic theory

Consider the recurrence relation

${\displaystyle x_{i+1}=4x_{i}(1-x_{i})\,}$

Then for almost every initial value x₀ in the unit interval [0,1],

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}}$

This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.

Physics

The number π appears routinely in equations describing fundamental principles of the universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.

• The cosmological constant:

${\displaystyle \Lambda ={{8\pi G} \over {3c^{2}}}\rho }$

• Heisenberg's uncertainty principle:

${\displaystyle \Delta x\Delta p\geq {\frac {h}{4\pi }}}$

• Einstein's field equation of general relativity:

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

• Coulomb's law for the electric force:

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

• Magnetic permeability of free space:

${\displaystyle \mu _{0}=4\pi \times 10^{-7}\,\mathrm {H/m} \,}$

Probability and statistics

In probability and statistics, there are many distributions whose formulæ contain π, including:

• probability density function (pdf) for the normal distribution with mean μ and standard deviation σ:

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

• pdf for the (standard) Cauchy distribution:

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

Note that since ${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1}$, for any pdf f(x), the above formulæ can be used to produce other integral formulæ for π.

A semi-interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using:

${\displaystyle \pi \approx {\frac {2nL}{xS}}}$

[As a practical matter, this approximation is poor and "converges" very slowly.]

Another approximation of π is to throw points randomly into a quarter of a circle with radius 1 that is inscribed in a square of length 1. Pi, the area of a unit circle, is then approximated as 4*(points in the quarter circle)/(total points).

History of π

Main article: History of Pi.

The value of π has been known in some form since antiquity. As early as the 20th century BC, Babylonian mathematicians were using π=25/8, which is within 0.5% of the exact value.

It is sometimes claimed that the Bible states that π=3, based on a passage in 1 Kings 7:23 giving measurements for a round basin. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. (Also, the basin may not have been exactly circular.)

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. [The approximation 355/113 is the very best one that may expressed with a three-digit numerator and denominator.]

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.

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.

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.

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.

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:

2 π = 6.2831853071795865

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.

The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 126 were correct [1] 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.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating pi to 707 decimal places. His routine was as follows: he would calculate new digits all morning; and then he would spend all afternoon checking his morning's work. His work was made possible by the recent invention of the logarithm and its tables by Napier and Briggs. This was the longest expansion of pi until the advent of the electronic digital computer centuries later. When these came around, it was found that Shanks had made a mistake somewhere in the 500s of decimal places, and that all succeeding digits were fallacious.

In the earliest years of the computer, the first expansion of pi to 1,000,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) near Washington, D.C. Dr. Shanks's son Oliver Shanks, also a mathematician, states that there is no connection to William Shanks, and in fact, the family's roots are in Central Europe.

Daniel Shanks and his team used two different power series for calculating the digital of pi. For one was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 1,000,000 digits of pi were published by the N.R.L.

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

${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}$

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

${\displaystyle (5+i)^{4}\cdot (-239+i)=-114244-114244i.}$

Formulæ of this kind are known as Machin-like formulae.

Many other expressions for π were developed and published by the incredibly-intuitive Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years.

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.

The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). 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 formulæ were used for this:

${\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}$

K. Takano (1982).

${\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}$

F. C. W. Störmer (1896).

These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.)

In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series:

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

This formula permits one to easily compute the k^th binary or hexadecimal digit of π, without 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).

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

${\displaystyle {\frac {\pi }{2}}=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=1+{\frac {1}{3}}\left(1+{\frac {2}{5}}\left(1+{\frac {3}{7}}\left(1+{\frac {4}{9}}(1+...)\right)\right)\right)}$

Newton.

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

Ramanujan.

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.

${\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}}$

David Chudnovsky and Gregory Chudnovsky.

${\displaystyle {\pi }=20\arctan {\frac {1}{7}}+8\arctan {\frac {3}{79}}}$

Euler.

Miscellaneous formulæ

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

${\displaystyle 3+{\frac {8}{60}}+{\frac {29}{60^{2}}}+{\frac {44}{60^{3}}}}$

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

• accurate to 9 digits:

${\displaystyle (63/25)((17+15{\sqrt {5}})/(7+15{\sqrt {5}}))}$

• accurate to 17 digits:

${\displaystyle 3+{\frac {48178703}{340262731}}}$

• accurate to 3 digits:

${\displaystyle {\sqrt {2}}+{\sqrt {3}}}$

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 special right triangles that are either isosceles or halves of equilateral triangles.

• The continued fraction representation of π can be used to generate successively better rational approximations of pi, which start off: 22/7, 333/106, 355/113. These approximations are the best possible rational approximations of π relative to the size of their denominators.

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:

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

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...)

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

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.)

The bill also recites Goodwin's previous accomplishments: "his solutions of the trisection of the angle, doubling the cube [and the value of π] 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.

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.

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

Open questions

The most pressing open question about π is whether it is a normal number -- whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulæ imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.

It is also unknown whether π and e are algebraically independent. However it is known that at least one of πe and π + e is transcendental (q.v.).

The nature of π

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulæ in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics.

For example, consider Coulomb's law (SI units)

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

Here, 4πr² is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as

${\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}$

and thus eliminate the need for π.

Fictional references

• Contact -- Carl Sagan's science fiction work. Sagan contemplates the possibility of finding a signature embedded in the base-11 expansion of Pi by the creators of the universe.
• Eon -- science fiction novel by Greg Bear. The protagonists measure the amount of space curvature using a device that computes π. Only in completely flat space/time will a circle have a circumference, diameter ratio of 3.14159... .
• Going Postal -- fantasy novel by Terry Pratchett. Famous inventor Bloody Stupid Johnson invents an organ/mail sorter that contains a wheel for which pi is exactly 3. This "New Pie" starts a chain of events that leads to the failure of the Ankh-Morpork Post Office (and possibly the destruction of the Universe all in one go.)
• π (film) -- On the relationship between numbers and nature: finding one without being a numerologist.
• The Simpsons -- "Pi is exactly 3!" was an announcement used by Professor Frink to gain the full attention of a hall full of scientists.
• Star Trek -- "Wolf in the Fold" -- When the computer of the Enterprise is taken over by an evil alien, Spock tells the computer to figure pi to the last digit, which forces the alien to leave.
• Time's Eye -- science fiction by Arthur C. Clarke and Stephen Baxter. In a world restructured by alien forces, a spherical device is observed whose circumference to diameter ratio appears to be an exact integer 3 across all planes. It is the first book in The Time Odyssey series.
• "Childhood's End" -- science-fiction novel by Arthur C. Clarke (1953). In the novel, a race of aliens that visits Earth mentions discovering a pattern within the digits of Pi. They indicate that they do not know the meaning of the pattern, but have devoted much effort to uncovering it. They speculate that the pattern could only have been placed there by the creator of the universe.
• Jimmy Neutron -- In revenge of the nanobots, Jimmy fixes the nanobots (or so he thinks) and they start vaporizing people because they think they are errors. He asks them to correct his paper (which is what they were made to do) and on it it says that pi is equal to 3. This causes them to explode and the world is saved.

π culture

March 14 (3/14 in US date format) marks Pi Day which is celebrated by many lovers of π.

On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π).

355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"

Singer Kate Bush's 2005 album "Aerial" contains a song titled "π," in which she sings π to its 137th decimal place; however, for an unknown reason, she omits the 79th to 100th decimal places.[2]

The band Hard 'n Phirm perform a song named π on their album Horses and Grasses. The song is 3 minutes, 14 seconds long. (Parts of) the song was used as the sound for a pi YTMND fad.

John Harrison (1693–1776) (of Longitude fame), devised a meantone temperament musical tuning system derived from π, now called Lucy Tuning.

Memorizing Pi

Ever since computers have calculated π to billions of decimal places, memorizing π has become a popular hobby for many people. The current world record is 83431 decimal places, and was set by a Japanese mental health counsellor named Akira Haraguchi, who is currently 59 years of age.[3] Before Haraguchi accomplished this on July 2, 2005, the world record was 42195, which was set by Hiroyuki Goto. See here, a website listing many people who have memorized impressive amounts of π.

There are many ways to memorize π, including the use of piems, which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics. Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this matter. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. See Pi mnemonics for examples.

The first thousand decimal places:

3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

See also

• List of topics related to pi
• Greek letter pi
• Calculus
• Geometry
• Trigonometric function
• Pi through experiment
• Proof that π is transcendental
• A simple proof that 22/7 exceeds π
• Feynman point
• Pi Day
• Lucy Tuning
• Cadaeic Cadenza
• Software for calculating π on personal computers

References

• Template:Journal reference

• Petr Beckmann, A History of Pi

External links

A more comprehensive approximation of Pi

• Pi to a million places
• Project Gutenberg E-Text containing a million digits of Pi
• Archives of Pi calculated to 1,000,000 or 10,000,000 places.
• Search the first 200 million digits of Pi for arbitrary strings of numbers
• Search π – search and print π's digits (up to 3.2 billion places)
• Statistics about the first 1.2 trillion digits of Pi

Calculation

• Calculating Pi: The open source project for calculating Pi.
• Stu's pi page, the fastest programs on the planet.
• Background Pi: An open source project for calculating Pi over many computers. (Inspired by "Calulating Pi", Above)
• PiFast: a fast program for calculating Pi with a large number of digits
• PiHex Project
• Super Pi: Another program to calculate Pi to the 33.55 millionth digit. Also used a benchmark
• PiSlice: A distributed computing project to calculate Pi
• Calculating the digits of π using generalised continued fractions - open source Python code

General

• J J O'Connor and E F Robertson: A history of Pi. Mac Tutor project
• A collection of Machin-type formulæ for Pi
• A proof that Pi Is Irrational
• PiFacts-Record Broken
• The Joy of Pi-About the Book
• From the Wolfram Mathematics site lots of formulæ for π
• Pi Symphony : An orchestral work by Lars Erickson based on the digits of pi and 'e'.
• PlanetMath: Pi
• The pi-hacks Yahoo! Group
• Finding the value of Pi
• Proof that Pi exists
• Friends of Pi Club (German and English)
• Determination of Pi at cut-the-knot
• LucyTuning - musical tuning derived from Pi
• The Pi Is Rational Page

Mnemonics

• One of the more popular mnemonic devices for remembering pi
• Andreas P. Hatzipolakis: PiPhilology. A site with hundreds of examples of π mnemonics
• Pi memorised as poetry
• First fifty digits of Pi, memorised as a humorous song
• Phrase to easily remember upto 8 decimal places of the value of Pi (See Item #3 on page)
• Online Pi Trainer to help memorise Pi
• Free software to help memorise Pi

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