The mathematical constant π is an irrational real number, approximately equal to 3.141592653..., which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with an Archimedes number) and as Ludolph's number.
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The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter.
The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery' and περίμετρος 'perimeter', i.e. 'circumference'.
π is Unicode character U+03C0 ("Greek small letter pi").
In Euclidean plane geometry, π is 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. The constant π may be defined in other ways that avoid the concepts of arc length and area, for example as twice the smallest positive x for which cos(x) = 0.^{[1]} The formulæ below illustrate other (equivalent) definitions.
The numerical value of π truncated to 50 decimal places is:
See the links below and those at sequence A00796 in OEIS for more digits.
While the value of pi has been computed to hundreds of millions of digits, practical science and engineering will rarely require more than 100 digits. As an example, computing the circumference of the Milky Way with a value of pi truncated at 40 digits would produce an error margin of less than the diameter of a proton. On the other hand, occasionally to produce accurate final results, some calculations may require more accurate intermediate values. Even so, a value of pi longer than a few hundred digits should never be necessary.^{[2]} The exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π.
Most formulas given for calculating the digits of π have desirable mathematical properties, but may be difficult to understand without a background in trigonometry and calculus. Nevertheless, it is possible to compute π using techniques involving only algebra and geometry.
For example:
This series is easy to understand, but is impractical in use as it converges to π very slowly. It requires more than 600 terms just to narrow its value to 3.14 (two places), and billions of terms to achieve accuracy to ten places.
One common classroom activity for experimentally measuring the value of π involves drawing a large circle on graph paper, then measuring its approximate area by counting the number of cells inside the circle. Since the area of the circle is known to be
π can be derived using algebra:
For a further explanation of this method as well as more computation methods see computing π.
The constant π 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.
Furthermore, 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 compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.
Often William Jones' book A New Introduction to Mathematics from 1706 is cited as the first text where the Greek letter π was used for this constant, but this notation became particularly popular after Leonhard Euler adopted it some years later (cf History of π).
The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = ^{25}⁄_{8}, which is within 0.5% of the true value.
The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.
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 as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim, which gives an approximate value of ~3.14; but it may suffice that the measurements are given in round numbers.
Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between ^{223}⁄_{71} and ^{22}⁄_{7}. The average of these two values is roughly 3.1419.
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 in the 5th century gave the approximation π = ^{62832}⁄_{20000} = 3.1416, correct when rounded off to four decimal places. He also acknowledged the fact that this was an approximation, which is quite advanced for the time period.
The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, ^{355}⁄_{113} and ^{22}⁄_{7}, in the 5th century.
The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of ^{π}⁄_{4} into the form
and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of ^{π}⁄_{4}, he was able to compute π to an accuracy of 13 decimal places.
The Persian astronomer Ghyath ad-din Jamshid Kashani (1350–1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:
By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.
In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and 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.
The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).
Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulæ for calculating π using elementary arithmetic invariably include notation such as "...", which indicates that the formula is really a formula for an infinite sequence of approximations to π. The more terms included in a calculation, the closer to π the result will get, but none of the results will be π exactly.
Consequently, 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 precision. 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} (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator.
The earliest numerical approximation of π is almost certainly the value 3. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.
All further improvements to the above mentioned "historical" approximations were done with the help of computers.
The constant π appears in many formulæ in geometry involving circles and spheres.
Geometrical shape | Formula |
---|---|
Circumference of circle of radius r and diameter d | |
Area of circle of radius r | |
Area of ellipse with semiaxes a and b | |
Volume of sphere of radius r and diameter d | |
Surface area of sphere of radius r and diameter d | |
Volume of cylinder of height h and radius r | |
Surface area of cylinder of height h and radius r | |
Volume of cone of height h and radius r | |
Surface area of cone of height h and radius r |
All of these formulae are a consequence of the formula for circumference. For example, the area of a circle of radius R can be accumulated by summing annuli of infinitesimal width using the integral . The others concern a surface or solid of revolution.
Also, the angle measure of 180° (degrees) is equal to π radians.
Many formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
Besides its simple continued-fraction representation [3; 7, 15, 1, 292, 1, 1, …], which displays no discernible pattern, π has many generalized continued-fraction representations generated by a simple rule, including these two.
(Other representations are available at The Wolfram Functions Site.)
Some results from number theory:
In the above three statements, "probability", "average", and "random" are taken in a limiting sense, i.e. we consider the probability for the set of integers {1, 2, 3,…, N}, and then take the limit as N approaches infinity.
The theory of elliptic curves and complex multiplication derives the approximation
which is valid to about 30 digits.
Consider the recurrence relation
Then for almost every initial value x_{0} in the unit interval [0,1],
This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.
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.
In probability and statistics, there are many distributions whose formulæ contain π, including:
Note that since , for any pdf f(x), the above formulæ can be used to produce other integral formulae 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:
[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. π, the area of a unit circle, is then approximated as 4*(points in the quarter circle) / (total points).
In the early years of the computer, the first expansion of π to 100,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 1961.
Daniel Shanks and his team used two different power series for calculating the digits of π. For one it 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 100,000 digits of π were published by the Naval Research Laboratory.
None of the formulæ given above can serve as an efficient way of approximating π. For fast calculations, one may use a formula 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
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.
The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The record as of 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:
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:
This formula permits one to fairly readily 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).
Fabrice Bellard claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of π [2]:
Other formulæ that have been used to compute estimates of π include:
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.
The base 60 representation of π, correct to eight significant figures (in base 10) is:
In addition, the following expressions approximate π:
Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. The current world record is 100,000 decimal places, set on October 3 2006 by Akira Haraguchi. [3] The previous record (83,431) was set by the same person on July 2 2005 [4], and the record previous to that (42,195) was held by Hiroyuki Goto.
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 (or: of course), 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 manner. 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. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of pi. Other methods include remembering "patterns" in the numbers (for instance, the "year" 1971 appears in the first fifty digits of pi).
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 formulae 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 (see Lindemann–Weierstrass theorem).
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)
Here, 4πr^{2} 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
and thus eliminate the need for π.
Some science-fiction stories posit location or situations in which the ratio of a circle's circumference to its diameter differs from π:
There are several examples in fiction of rogue AIs being crashed by inviting them to contemplate π. It is often not clear why π is better for such uses than other irrational numbers such as e or .
Finally there are many cases in fiction where numbers that a priori have nothing to do with geometry or analysis are chosen to include the first few digits of π. These references range from conspicuous jokes to easter eggs.