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 The minuscule, or lower-case, pi The mathematical constant π (written as "pi" when the Greek letter is not available) is ubiquitous in mathematics and physics. In Euclidean plane geometry, π may be defined as either the ratio of a circle's circumference to its diameter, or as the area of a circle of radius 1 (the unit circle). Most modern textbooks define π analytically using trigonometric functions, e.g. 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. Pi is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number. The numerical value of π approximated to 64 decimal places (sequence A000796 in OEIS) is:
More digits of π are also available. See pi to 1,000 places, 10,000 places, 100,000 places, and 1,000,000 places.
PropertiesPi is an irrational number: that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In fact, the number is transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational (equivalently, integer) coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. This means that it is impossible to express π using only a finite number of integers, fractions and their square roots. This result establishes the impossibility of squaring the circle: it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The reason is that the coordinates of all points that can be constructed with ruler and compass are constructible numbers. While the original Greek letter for pi was phonetically equivalent to the English letter p, it has now evolved to be pronounced like the word pie in most circles. Formulas involving πGeometryPi appears in many formulas in geometry involving circles and spheres.
AnalysisMany formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
\frac{\sqrt2}2 \frac{\sqrt{2+\sqrt2}}2 \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots<math>
Complex analysis
Continued fractionsPi has many continued fractions representations, including:
(You can see other representations at The Wolfram Functions Site (http://functions.wolfram.com/Constants/Pi/10/).) Number theorySome results from number theory:
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. Dynamical systems / ergodic theoryIn dynamical systems theory (see also ergodic theory), for almost every real-valued x0 in the interval [0,1],
where the xi are iterates of the Logistic map for r = 4. PhysicsFormulas from physics.
Probability and statisticsIn probability and statistics, there are many distributions whose formulas contain π, including:
Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1<math>, for any pdf f(x), the above formulas can be used to produce other integral formulas for π. An 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:
HistoryThe symbol "π" for Archimedes' constant was first introduced in 1706 by William Jones when he published A New Introduction to Mathematics, although the same symbol had been used earlier to indicate the circumference of a circle. The notation became standard after it was adopted by Leonhard Euler. In either case, π is the first letter of περιμετρος (perimetros), meaning 'measure around' in Greek. Here is a brief chronology of π:
Numerical approximations of πDue to the transcendental nature of π, there are no nice closed expressions for π. Therefore numerical calculations must use approximations to the number. 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. An Egyptian scribe named Ahmes is the source of the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes states 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 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 radius 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 digit as:
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 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today. None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as Machin's:
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
Formulas of this kind are known as Machin-like formulas. 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) stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulas were used for this:
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 1996 David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series:
\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 kth binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website (http://www.nersc.gov/~dhbailey/) 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 formulas that have been used to compute estimates of π include:
\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>
Open questionsThe most pressing open question about π is whether it is a normal number, i.e. 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 2000 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 conjecture of chaos theory. See Bailey's above mentioned web site for details. It is also unknown whether π and e are algebraically independent, i.e. whether there is a polynomial relation between π and e with rational coefficients. The nature of πIn 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 formulae 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. The reason it occurs so often in physics is simply because it's convenient in many physical models. For example Coulomb's law <math> F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2} <math>, here <math>4 \pi r^2<math> is just the surface area of sphere of radius r, which 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 <math> F = \frac{q_1 q_2}{r^2} <math> and thus eliminate the need for π. π cultureThere is an entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, which is known as piphilology. For example, part of the school cheer of MIT is: "Cosine, secant, tangent, sine! 3 point 1 4 1 5 9!" See Pi mnemonics for more examples. March 14 (3/14) marks Pi Day which is celebrated by many lovers of π. On July 22, Pi Approximation Day is celebrated (22/7 is a popular approximation of π). Furthermore, many talk of "pi o clock" [fifteen seconds past fourteen minutes past 3 (3:14:15) is slightly less than pi o clock]. Another example of math-humor is this approximation of π: Take the number "1234", transpose the first two digits and the last two digits, so the number becomes "2143". Divide that number by "two-two" (22, so 2143/22 = 97.40909...). Take the two-squaredth root (4th root) of this number. The final outcome is remarkably close to π: 3.14159265. Related articles
External linksDigit resources
Calculation
General
Mnemonics
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