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| Yum! I like pie ^-^ | |||
| {{otheruses|pi}} | |||
| ], or lower-case, pi]] | |||
| The ] '''π''' represents the ] of a ]'s ] to its ] and is commonly used in ], ], and ]. π is the lowercase Greek letter equivalent to "p" in the ]; its name is "pi" (pronounced ''pie'' in English, but ''pee'' in Greek), and this spelling can be used in typographical contexts where the ] is not available. π is also known as ''']' constant''' (not to be confused with ]) and ''']'s number'''. | |||
| In ], π may be defined either as the ] of a ]'s ] to its ], or as the ratio of a circle's ] to the area of a square whose side is the radius. Advanced textbooks define π ] using ]s, for example as the smallest positive ''x'' for which ](''x'') = 0, or as twice the smallest positive ''x'' for which ](''x'') = 0. | |||
| All these definitions are equivalent. | |||
| The numerical value of π approximated to 100 ] {{OEIS|id=A000796}} is: | |||
| :3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 | |||
| Although this precision is more than sufficient for use in ] and ], much effort over the last few centuries has been put into computing as many digits as possible, for prestige as well as mathematical inquiry. Digits of π are available from multiple resources on the Internet, and a regular ] can be used to compute billions of digits. Recent calculations with ] have found over 1 trillion digits of π, but despite this, no pattern in the digits has ever been found; in one sense, this is not surprising; ] ]s are ], and almost all have no ] pattern. | |||
| == Properties == | |||
| π is an ]; that is, it cannot be written as the ratio of two ]s, as was proven in ] by ]. | |||
| π is also ], as was proven by ] in ]. This means that there is no ] with ] (equivalently, ]) coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not ]. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to ], that is, it is impossible to construct, using ] alone, a square whose area is equal to the area of a given circle. | |||
| == Formulae involving π == | |||
| ===Geometry=== | |||
| <math>\pi</math> appears in many formulae in ] involving ]s and ]s. | |||
| {| border="1" cellspacing="4" cellpadding="4" style="border-collapse: collapse;" | |||
| !Geometrical shape | |||
| !Formula | |||
| |- | |||
| |] of circle of ] ''r'' and ] ''d'' | |||
| |<math>C = \pi d = 2 \pi r \,\!</math> | |||
| |- | |||
| |] of circle of radius ''r'' | |||
| |<math>A = \pi r^2 \,\!</math> | |||
| |- | |||
| |Area of ] with semiaxes ''a'' and ''b'' | |||
| |<math>A = \pi a b \,\!</math> | |||
| |- | |||
| |] of sphere of radius ''r'' and diameter ''d'' | |||
| |<math>V = \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3 \,\!</math> | |||
| |- | |||
| |] of sphere of radius ''r'' | |||
| |<math>A = 4 \pi r^2 \,\!</math> | |||
| |- | |||
| |Volume of ] of height ''h'' and radius ''r'' | |||
| |<math>V = \pi r^2 h \,\!</math> | |||
| |- | |||
| |Surface area of cylinder of height ''h'' and radius ''r'' | |||
| |<math>A = 2 ( \pi r^2 ) + ( 2 \pi r ) h = 2 \pi r (r + h) \,\!</math> | |||
| |- | |||
| |Volume of ] of height ''h'' and radius ''r'' | |||
| |<math>V = \frac{1}{3} \pi r^2 h \,\!</math> | |||
| |- | |||
| |Surface area of cone of height ''h'' and radius ''r'' | |||
| |<math>A = \pi r \sqrt{r^2 + h^2} + \pi r^2 = \pi r (r + \sqrt{r^2 + h^2}) \,\!</math> | |||
| |} | |||
| (All of these are a consequence of the first one, as the area of a circle can be written as | |||
| ''A'' = ∫(2''πr'')d''r'' ("sum of ] of infinitesimal width"), and others concern a surface or ].) | |||
| Also, the ] measure of 180° (]) is equal to π ]s. | |||
| ===Analysis=== | |||
| Many formulae in ] contain π, including ] (and ]) representations, ]s, and so-called ]. | |||
| *], ] (]): | |||
| :<math>\frac2\pi= | |||
| \frac{\sqrt2}2 | |||
| \frac{\sqrt{2+\sqrt2}}2 | |||
| \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots</math> | |||
| *]' formula (]): | |||
| :<math>\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4}</math> | |||
| :This commonly cited infinite series is usually written as above, but is more technically expressed as: | |||
| :<math>\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}</math> | |||
| *] (see that article for a proof): | |||
| :<math> \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} </math> | |||
| : <math>\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}</math> | |||
| *1995 Bailey-Borwein-Plouffe algorithm | |||
| :<math>\pi=\sum_{k=0}^\infty\frac{1}{16^k}\left </math> | |||
| *An ] formula from ] (see also ] and ]): | |||
| :<math>\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</math> | |||
| *], first solved by ] (see also ]): | |||
| :<math>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math> | |||
| :<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}</math> | |||
| :and generally, <math>\zeta(2n)</math> is a rational multiple of <math>\pi^{2n}</math> for positive integer n | |||
| *] evaluated at 1/2: | |||
| :<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}</math> | |||
| *]: | |||
| :<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> | |||
| *] (called by ] "the most remarkable formula in mathematics"): | |||
| :<math>e^{i \pi} + 1 = 0\;</math> | |||
| *Property of ] (see also ]): | |||
| :<math>\sum_{k=1}^{n} \phi (k) \sim 3 n^2 / \pi^2</math> | |||
| *Area of one quarter of the unit circle: | |||
| :<math>\int_0^1 \sqrt{1-x^2}\,dx = {\pi \over 4}</math> | |||
| *An application of the ] | |||
| :<math>\oint\frac{dz}{z}=2\pi i ,</math> | |||
| :where the path of integration is a circle around the origin, traversed in the standard (anti-clockwise) direction. | |||
| ===Continued fractions=== | |||
| π has many ]s representations, including: | |||
| :<math> \frac{4}{\pi} = 1 + \frac{1}{3 + \frac{4}{5 + \frac{9}{7 + \frac{16}{9 + \frac{25}{11 + \frac{36}{13 + ...}}}}}} </math> | |||
| (Other representations are available at .) | |||
| ===Number theory=== | |||
| Some results from ]: | |||
| *The ] that two ]ly chosen integers are ] is 6/π<sup>2</sup>. | |||
| *The probability that a randomly chosen integer is ] is 6/π<sup>2</sup>. | |||
| *The ] number of ways to write a positive integer as the sum of two ]s (order matters) is π/4. | |||
| * The ] of (1-1/p<sup>2</sup>) over the primes, ''p'', is 6/π<sup>2</sup>.<math> \prod_{p\in\mathbb{P}} \left(1-\frac {1} {p^2} \right) = \frac {6} {\pi^2} </math> | |||
| 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 ] as ''N'' approaches infinity. | |||
| The remarkable fact that | |||
| : <math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...</math> | |||
| or equivalently, | |||
| : <math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007...</math> | |||
| can be explained by the theory of ]. | |||
| ===Dynamical systems and ergodic theory=== | |||
| Consider the ] | |||
| :<math>x_{i+1} = 4 x_i (1 - x_i) \,</math> | |||
| Then for ] initial value ''x''<sub>0</sub> in the ] , | |||
| :<math> \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \sqrt{x_i} = \frac{2}{\pi} </math> | |||
| This recurrence relation is the ] with parameter ''r'' = 4, known from ]s theory. See also: ]. | |||
| ===Physics=== | |||
| In ], appearance of π in formulae is usually only a matter of convention and normalization. For example, by using the reduced ] <math> \hbar = \frac{h}{2\pi} </math> one can avoid writing π explicitly in many formulae of quantum mechanics. In fact, the reduced version is the more fundamental, and presence of factor ''1/2π'' in formulas using ''h'' can be considered an artifact of the conventional definition of Planck's constant. | |||
| *]: | |||
| :<math> \Delta x \Delta p \ge \frac{h}{4\pi} </math> | |||
| *] of ]: | |||
| :<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math> | |||
| *] for the ]: | |||
| :<math> F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2} </math> | |||
| *]: | |||
| :<math> \mu_0 = 4 \pi \times 10^{-7}\,\mathrm{H/m}\,</math> | |||
| ===Probability and statistics=== | |||
| In ] and ], there are many ] whose formulae contain π, including: | |||
| *] (pdf) for the ] with ] μ and ] σ: | |||
| :<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math> | |||
| *pdf for the (standard) ]: | |||
| :<math>f(x) = \frac{1}{\pi (1 + x^2)}</math> | |||
| Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math>, for any pdf ''f''(''x''), the above formulae can be used to produce other integral formulas for π. | |||
| An interesting empirical approximation of π is based on ] 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: | |||
| :<math>\pi \approx \frac{2nL}{xS}</math> | |||
| == History of π == | |||
| ''Main article: ]''. | |||
| π has been known in some form since antiquity. References to measurements of a circular basin in the ] give a corresponding value of 3 for π: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." — ] 7:23; KJV. (Nehemiah, a ] Jewish rabbi and mathematician explained this apparent lack of precision in π, by considering the thickness of the basin, and assuming that the thirty cubits was the inner circumference, while the ten cubits was the diameter of the outside of the basin.) | |||
| == 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 ]s of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 ]) 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 ] expansion of π. | |||
| An Egyptian scribe named ] wrote the oldest known text to give an approximate value for π. The ] dates from the ] ]—though Ahmes stated that he copied a ] ]—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 ] computed π to 3.141014 (good to three decimal places) in AD ] and suggested that 3.14 was a good approximation. | |||
| The Indian mathematician and astronomer ] 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 ] computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the ]. | |||
| The Iranian mathematician and astronomer, ], 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 ] (''circa'' ]) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his ]. | |||
| The Slovene mathematician ] in ] calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until ], when ] calculated 208 decimal places of which the first 152 were correct. Vega improved ]'s formula from ] and his method is still mentioned today. | |||
| None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as ]: | |||
| : <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} </math> | |||
| together with the ] expansion of the function ](''x''). This formula is most easily verified using ] of ]s, starting with | |||
| :<math>(5+i)^4\cdot(-239+i)=-114244-114244i.</math> | |||
| Formulas of this kind are known as ''Machin-like formulas''. | |||
| Extremely long decimal expansions of π are typically computed with the ] and ]; the ] which was invented in ] has also been used in the past. | |||
| The first one million digits of π and 1/π are available from ] (see external links below). | |||
| The current record (December ]) by ] of ] stands at 1,241,100,000,000 digits, which were computed in September ] on a 64-node ] ] 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: | |||
| :<math> \frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}</math> | |||
| :K. Takano (]). | |||
| : <math> \frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}</math> | |||
| :F. C. W. Störmer (]). | |||
| 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. | |||
| In ] ], together with ] and ], discovered a new formula for π as an ]: | |||
| : <math>\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)</math> | |||
| This formula permits one to easily compute the ''k''<sup>th</sup> ] or ] digit of π, without | |||
| having to compute the preceding ''k'' − 1 digits. contains the derivation as well as implementations in various ]. The ] project computed 64-bits around the ]th bit of π (which turns out to be 0). | |||
| Other formulas that have been used to compute estimates of π include: | |||
| :<math> | |||
| \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) | |||
| </math> | |||
| :]. | |||
| :<math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} </math> | |||
| :]. | |||
| 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. | |||
| :<math> \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}} </math> | |||
| :] and ]. | |||
| : <math>{\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79} </math> | |||
| :]. | |||
| On computers running ] ], the program ] 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. | |||
| ===Miscellaneous formulas=== | |||
| In ] 60, π can be approximated to eight significant figures as | |||
| :<math> 3 + \frac{8}{60} + \frac{29}{60^2} + \frac{44}{60^3}</math> | |||
| In addition, the following expressions can be used to estimate π | |||
| * accurate to 9 digits: | |||
| :<math>(63/25)((17+15\sqrt 5)/(7+15\sqrt5))</math> | |||
| * accurate to 17 digits: | |||
| :<math>3 + \frac{48178703}{340262731}</math> | |||
| * accurate to 3 digits: | |||
| :<math>\sqrt{2} + \sqrt{3}</math> | |||
| :] conjectured that ] 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 ]s which are either ] or halves of ] triangles. | |||
| ===Less accurate approximations=== | |||
| In 1897, a physician and amateur mathematician from ] named ] believed that the ] value of π was wrong. He proposed a bill to Indiana Representative ] 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 ], ] having been already accepted as contributions to science by the ]....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 ]. The claims trisection of an angle and the doubling of the cube are particularly widespread in crank literature. | |||
| The Indiana ] referred the bill to the Committee on Swamp Lands, which ] 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 ] arrived in ] to secure the annual appropriation for the ]. 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. | |||
| == Open questions == | |||
| The most pressing open question about π is whether it is a ] -- 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". This must be true in any base, not just in base 10. Current knowledge in this direction 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 ] that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible ] of ]. See Bailey's above mentioned web site for details. | |||
| It is also unknown whether π and ] are ], i.e. whether there is a polynomial relation between π and ''e'' with rational coefficients. | |||
| ], (1693-1776) (of Longitude fame), devised a ] musical tuning system derived from π. This ] system (due to the unique mathematical properties of π), can map all musical intervals, harmony and harmonics. This suggests that musical harmonics beat, and that using π could provide a more precise model for the analysis of both musical and other harmonics in vibrating systems. | |||
| == The nature of π == | |||
| In ] the sum of the angles of a ] may be more or less than π ], 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 formulae in which π appears. So, in particular, π is not affected by the ]; it is not a ] but a mathematical constant defined independently of any physical measurements. The reason it occurs so often in physics is simply because it's convenient in many physical models. | |||
| For example, consider ] | |||
| :<math> F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2} </math>. | |||
| Here, 4''πr''<sup>2</sup> 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 ] is used, it can be written as | |||
| :<math> F = \frac{q_1 q_2}{r^2} </math> | |||
| and thus eliminate the need for π. | |||
| == Fictional references == | |||
| * '']'' -- ] ] work. Sagan contemplates the possibility of finding a signature embedded in the ] expansion of Pi by the creators of the universe. | |||
| * '']'' -- On the relationship between numbers and nature: finding one without being a ]. | |||
| * '']'' -- ] by ] and ]. 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. T | |||
| * '']'' -- "Pi equals exactly 3" was an announcement used by ] to gain the full attention of a hall full of scientists. | |||
| == π culture == | |||
| There is an entire field of humorous yet serious study that involves the use of ]s to remember the digits of π, which is known as ]. See ] for examples. | |||
| ] (3/14 in ] date format) marks ] which is celebrated by many lovers of π. | |||
| On ], ] is celebrated (22/7 - in European date format - is a popular approximation of π). | |||
| In the early hours of Saturday ], ], a Japanese mental health counsellor, Akira Haraguchi, 59, managed to recite Pi's first 83,431 decimal places from ], thus breaking the standing world record . | |||
| 355/113 (3.1415929) is sometimes jokingly referred to as "Not Pi, but an incredible simulation!" | |||
| == Related articles == | |||
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| == External links == | |||
| ===Digit resources=== | |||
| * | |||
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| * – search and print π's digits (up to 3.2 billion places) | |||
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| ===Calculation=== | |||
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| *] - open source ] code | |||
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Revision as of 03:12, 15 September 2005
Yum! I like pie ^-^