Pi
The number π (; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry[a] as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.[1] It is also referred to as Archimedes's constant.[2][3]
As an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.
It is known that π is a transcendental number:[2] It is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later, when the Madhava–Leibniz series was discovered by the Kerala school of astronomy and mathematics, documented in the Yuktibhāṣā, written around 1530.[4]
The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits.[6][7] The primary motivation for these computations is as a test case to develop efficient algorithms to calculate numeric series, as well as the quest to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. In more modern mathematical analysis, it is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with the geometry of circles, such as number theory and statistics, and in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of the scientific community. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called
π.
Because
π is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge.
The constant
π is represented in this mosaic outside the Mathematics Building at the Technical University of Berlin.
The association between imaginary powers of the number
e and points on the unit circle centered at the origin in the complex plane given by Euler's formula
π can be estimated by computing the perimeters of circumscribed and inscribed polygons.
Archimedes developed the polygonal approach to approximating
π.
Comparison of the convergence of several historical infinite series for
π.
Sn is the approximation after taking
n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)
Isaac Newton used infinite series to compute
π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
[77] The earliest known use of the Greek letter
π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706
[1] Leonhard Euler popularized the use of the Greek letter
π in works he published in 1736 and 1748.
John von Neumann was part of the team that first used a digital computer, ENIAC, to compute
π.
As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of
π increased dramatically. The vertical scale is logarithmic.
Srinivasa Ramanujan, working in isolation in India, produced many innovative series for computing
π.
Random dots are placed on the quadrant of a square with a circle inscribed in it.
Monte Carlo methods, based on random trials, can be used to approximate π.
Five random walks with 200 steps. The sample mean of
|W200| is
μ = 56/5, and so
2(200)μ−2 ≈ 3.19 is within
0.05 of
π.
The area of the circle equals
π times the shaded area. The area of the unit circle is
π.
Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.
Sine and cosine functions repeat with period 2
π.
The overtones of a vibrating string are eigenfunctions of the second derivative, and form a harmonic progression. The associated eigenvalues form the arithmetic progression of integer multiples of
π.
The ancient city of Carthage was the solution to an isoperimetric problem, according to a legend recounted by Lord Kelvin (Thompson 1894): those lands bordering the sea that Queen Dido could enclose on all other sides within a single given oxhide, cut into strips.
An animation of a geodesic in the Heisenberg group, showing the close connection between the Heisenberg group, isoperimetry, and the constant
π. The cumulative height of the geodesic is equal to the area of the shaded portion of the unit circle, while the arc length (in the Carnot–Carathéodory metric) is equal to the circumference.
A graph of the Gaussian function
ƒ(x) = e−x2. The coloured region between the function and the
x-axis has area
√π.
π can be computed from the distribution of zeros of a one-dimensional Wiener process
Uniformization of the Klein quartic, a surface of genus three and Euler characteristic −4, as a quotient of the hyperbolic plane by the symmetry group PSL(2,7) of the Fano plane. The hyperbolic area of a fundamental domain is
8π, by Gauss–Bonnet.
The techniques of vector calculus can be understood in terms of decompositions into spherical harmonics (shown)
Einstein's equation states that the curvature of spacetime is produced by the matter–energy content.
Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles. Here illustrated is the complex logarithm of the Gamma function.
The Hopf fibration of the 3-sphere, by Villarceau circles, over the complex projective line with its Fubini–Study metric (three parallels are shown). The identity
S3(1)/S2(1) = π/2 is a consequence.
Each prime has an associated Prüfer group, which are arithmetic localizations of the circle. The L-functions of analytic number theory are also localized in each prime
p.
Solution of the Basel problem using the Weil conjecture: the value of
ζ(2) is the hyperbolic area of a fundamental domain of the modular group, times
π/2.
π appears in characters of p-adic numbers (shown), which are elements of a Prüfer group. Tate's thesis makes heavy use of this machinery.
[196] Theta functions transform under the lattice of periods of an elliptic curve.
The Witch of Agnesi, named for Maria Agnesi (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.
The Cauchy distribution governs the passage of Brownian particles through a membrane.
The Mandelbrot set can be used to approximate
π.
A pi pie. Pies are circular, and "pie" and
π are homophones, making pie a frequent subject of pi puns.