Pi | Read 10 Min

A unit circle is rolled across a surface; after a full revolution, the center of the circle has moved π units forwards. — An illustration of the value of pi

The number $π$ (/paɪ/ ^ⓘ; spelled out as pi) 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 $π$ , to avoid relying on the definition of the length of a curve.

The number $π$ is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as $⁠ 22 / 7 ⁠$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter 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 $π$ implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of $π$ appear to be evenly distributed, but no proof of this conjecture has been found.

For thousands of years, mathematicians have attempted to extend their understanding of $π$ , sometimes by computing its value to a high degree of accuracy. 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 mathematicians approximated $π$ to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for $π$ , 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 $π$ , 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 hundreds of 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, $π$ 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 $π$ makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to $π$ have been published, and record-setting calculations of the digits of $π$ 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 $π$ , sometimes spelled out as pi. In English, $π$ is pronounced as "pie" (/paɪ/ PY). In mathematical use, the lowercase letter $π$ is distinguished from its capitalized and enlarged counterpart $Π$ , which denotes a product of a sequence, analogously to how $Σ$ denotes summation.

The choice of the symbol $π$ is discussed in the section § Adoption of the symbol $π$ .

Definition

$π$ is commonly defined as the ratio of a circle's circumference $C$ to its diameter $d$ : $\pi ={\frac {C}{d}}$ In Euclidean geometry, 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 .

Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined using limits—a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation , as the integral:

An integral such as this was proposed as a definition of $π$ by Karl Weierstrass, who defined it directly as an integral in 1841.

Integration is no longer commonly used in a first analytical definition because, as Remmert 2012 explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of $π$ that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: $π$ is twice the smallest positive number at which the cosine function equals 0. $π$ is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation.

In a similar spirit, $π$ can be defined using properties of the complex exponential, $exp z$ , of a complex variable $z$ . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which $exp z$ is equal to one is then an (imaginary) arithmetic progression of the form: and there is a unique positive real number $π$ with this property.

A variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex numbers of absolute value one. The number $π$ is then defined as half the magnitude of the derivative of this homomorphism.

Irrationality and normality

$π$ is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as $⁠ 22 / 7 ⁠$ and $⁠ 355 / 113 ⁠$ are commonly used to approximate $π$ , but no common fraction (ratio of whole numbers) can be its exact value. Because $π$ 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 $π$ is irrational; they are generally proofs by contradiction and require calculus. The degree to which $π$ can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of $e$ but smaller than the measure of Liouville numbers.

The digits of $π$ 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 (of any given length) appear equally often. The conjecture that $π$ is normal has not been proven or disproven.

Since the advent of computers, a large number of digits of $π$ have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of $π$ , 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 $π$ '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 $π$ .

Transcendence

In addition to being irrational, $π$ 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 $e$ .

The transcendence of $π$ has two important consequences: First, $π$ cannot be expressed using any finite combination of rational numbers and square roots or nth roots (such as or ). 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 $π$ and $e$ are algebraically independent ("relatively transcendental"). This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.

Continued fractions

As an irrational number, $π$ cannot be represented as a common fraction. However, like every number, $π$ 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 $π$ ; the first four of these are $3$ , $⁠ 22 / 7 ⁠$ , $⁠ 333 / 106 ⁠$ , and $⁠ 355 / 113 ⁠$ . 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 $π$ than any other fraction with the same or a smaller denominator. Because $π$ is transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, $π$ cannot have a periodic continued fraction. Although the simple continued fraction for $π$ (with numerators all 1, shown above) 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 (in order of increasing accuracy) ⁠22/7⁠, ⁠333/106⁠, ⁠355/113⁠, ⁠52163/16604⁠, ⁠103993/33102⁠, ⁠104348/33215⁠, and ⁠245850922/78256779⁠. (List is selected terms from OEIS: A063674 and OEIS: A063673.)
Digits: The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510... (see OEIS: A000796)

Digits in other number systems

The first 48 binary (base 2) digits (called bits) are 11.001001000011111101101010100010001000010110100011... (see OEIS: A004601)
The first 36 digits in ternary (base 3) are 10.010211012222010211002111110221222220... (see OEIS: A004602)
The first 20 digits in hexadecimal (base 16) are 3.243F6A8885A308D31319... (see OEIS: A062964)
The first five sexagesimal (base 60) digits are 3;8,29,44,0,47 (see OEIS: A060707)

Complex numbers and Euler's identity

Any complex number, say $z$ , can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or $r$ ) is used to represent $z$ 's distance from the origin of the complex plane, and the other (angle or $φ$ ) the counter-clockwise rotation from the positive real line:

where $i$ is the imaginary unit satisfying . The frequent appearance of $π$ 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 $e$ is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of $e$ 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 $n$ different complex numbers $z$ satisfying ⁠⁠, and these are called the " $n$ th roots of unity" and are given by the formula:

History

Surviving approximations of $π$ prior to the 2nd century CE are accurate to one or two decimal places at best. The earliest written approximations are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BCE has a geometrical statement that, by implication, treats $π$ as ⁠25/8⁠ = 3.125. In Egypt, the Rhind Papyrus, dated around 1650 BCE but copied from a document dated to 1850 BCE, has a formula for the area of a circle that treats $π$ as . Although some pyramidologists have theorized that the Great Pyramid of Giza was built with proportions related to $π$ , this theory is not widely accepted by scholars. In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the 1st or 2nd millennium BCE, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.