Pi


The number is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining, to avoid relying on the definition of the length of a curve.
The number is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of appear to be randomly distributed, but no proof of this conjecture has been found.
For thousands of years, mathematicians have attempted to extend their understanding of, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250BC, the Greek mathematician Archimedes created an algorithm to approximate with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for, based on infinite series, was discovered a millennium later. 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. The invention of calculus soon led to the calculation of hundreds of digits of, 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 to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors.
Because it relates to a circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas having little to do with geometry, such as number theory and statistics, and in modern mathematical analysis can be defined without any reference to geometry. The ubiquity of makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to have been published, and record-setting calculations of the digits of often result in news headlines.

Fundamentals

Name

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter, sometimes spelled out as pi. In English, is pronounced as "pie". In mathematical use, the lowercase letter is distinguished from its capitalized and enlarged counterpart, which denotes a product of a sequence, analogously to how denotes summation.
The choice of the symbol is discussed in the section § Adoption of the symbol.

Definition

is commonly defined as the ratio of a circle's circumference to its diameter :
The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio.
In modern mathematics, this definition is not fully satisfactory for several reasons. Firstly, it lacks a rigorous definition of the length of a curved line. Such a definition requires at least the concept of a limit, or, more generally, the concepts of derivatives and integrals. Also, diameters, circles and circumferences can be defined in Non-Euclidean geometries, but, in such a geometry, the ratio need not to be a constant, and need not to equal to. Also, there are many occurrences of in many branches of mathematics that are completely independent from geometry, and in modern mathematics, the trend is to built geometry from algebra and analysis rather than independently from the other branches of mathematics. For these reasons, the following characterizations can be taken as definitions of :
  • is the smallest positive zero of the sine function; that is, and is the smallest positive number with this property.
  • is the smallest positive difference between two zeros of,, and .
The sine and the cosine satisfy the differential equation, and all solutions of this equation are periodic. This leads to the conceptual definition:
is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as and are commonly used to approximate, but no common fraction can be its exact value. Because is irrational, it has an infinite number of digits in its decimal representation, and does not settle into an infinitely repeating pattern of digits. There are several proofs that is irrational; they are generally proofs by contradiction and require calculus. The degree to which can be approximated by rational numbers is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of but smaller than the measure of Liouville numbers.
The digits of have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits appear equally often. The conjecture that is normal has not been proven or disproven.
Since the advent of computers, a large number of digits of have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of 's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of. This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.

Transcendence

In addition to being irrational, is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as. This follows from the so-called Lindemann–Weierstrass theorem, which also establishes the transcendence of the constant '.
The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or nth roots. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.
An unsolved problem thus far is the question of whether or not the numbers ' and are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.

Continued fractions

As an irrational number, cannot be represented as a common fraction. But every number, including, can be represented by an infinite series of nested fractions, called a simple continued fraction:
Truncating the continued fraction at any point yields a rational approximation for ; the first four of these are,,, and. These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to than any other fraction with the same or a smaller denominator. Because is transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, cannot have a periodic continued fraction. Although the simple continued fraction for also does not exhibit any other obvious pattern, several non-simple continued fractions do, such as:

Approximate value and digits

Some approximations of pi include:
  • Integers: 3
  • Fractions: Approximate fractions include ,,,,,, and.
  • Digits: The first 50 decimal digits are
Digits in other number systems
Any complex number, say, can be expressed using a pair of real numbers. In the polar coordinate system, one number is used to represent 's distance from the origin of the complex plane, and the other the counter-clockwise rotation from the positive real line:
where is the imaginary unit satisfying. The frequent appearance of in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula:
where the constant is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of and points on the unit circle centred at the origin of the complex plane. Setting in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants:
There are different complex numbers satisfying, and these are called the "th roots of unity" and are given by the formula: