The number (; 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 , 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 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 implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of appear to be evenly distributed, but no proof of this conjecture has been found.
Mathematicians have long 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 development of calculus in the 17th century 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 extended the decimal representation of to hundreds of trillions of digits.
Because it relates to a circle, is found in formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is found as well in formulae from cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas such as number theory, statistics, and in modern mathematical analysis is ubiquitous.
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.
Definition
is commonly defined as the ratio of a circle's circumference to its diameter :
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 limitsa 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 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 explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: is twice the smallest positive number at which the cosine function equals 0. 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, can be defined using properties of the complex exponential, , of a complex variable . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which is equal to one is then an (imaginary) arithmetic progression of the form:
and there is a unique positive real number with this property.
Irrationality and normality 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 (ratio of whole numbers) 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 (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 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 (of any given length) 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 .
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 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 th 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 successdespite the fact that it is mathematically impossible. , p. 185.
An unsolved problem thus far is the question of whether or not the numbers and 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, cannot be represented as a common fraction. However, like every number, 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 (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) , , , , , , and . (List is selected terms from and .) Digits: The first 50 decimal digits are (see )
Digits in other number systems The first 48 binary (base 2) digits (called bits) are (see ) The first 36 digits in ternary (base 3) are (see ) The first 20 digits in hexadecimal (base 16) are (see ) The first five sexagesimal (base 60) digits are 3;8,29,44,0,47 Ptolemy used a three-sexagesimal-digit approximation, and Jamshīd al-Kāshī expanded this to nine digits; see (see )
Complex numbers and Euler's identity
Any complex number, say , can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or ) is used to represent 's distance from the origin of the complex plane, and the other (angle or ) 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:
History
Surviving approximations of prior to the 2nd centuryCE 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 1900–1600BCE has a geometrical statement that, by implication, treats as . In Egypt, the Rhind Papyrus, dated around 1650BCE but copied from a document dated to 1850BCE, has a formula for the area of a circle that treats as . Although some pyramidologists have theorized that the Great Pyramid of Giza was built with proportions related to , 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 millenniumBCE, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.
Polygon approximation era