The number π ( ; 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. 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 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: π 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 π, sometimes spelled out as pi. In English, π is pronounced as "pie" ( 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:
π = C d . {\displaystyle \pi ={\frac {C}{d}}.}
In Euclidean geometry, the ratio C d {\textstyle {\frac {C}{d}}} 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 C d {\textstyle {\frac {C}{d}}} . 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 x 2 + y 2 = 1 {\textstyle x^{2}+y^{2}=1} , as the integral:
π = ∫ − 1 1 d x 1 − x 2 . {\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}
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:
{ … , − 2 π i , 0 , 2 π i , 4 π i , … } = { 2 π k i ∣ k ∈ Z } {\displaystyle \{\dots ,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}}
and there is a unique positive real number π with this property.
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 x 5 120 − x 3 6 + x = 0 {\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0} . 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 31 3 {\displaystyle {\sqrt[{3}]{31}}} or 10 {\displaystyle {\sqrt {10}}} ). 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:
π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}
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:
z = r ⋅ ( cos φ + i sin φ ) , {\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ),}
where i is the imaginary unit satisfying i 2 = − 1 {\displaystyle i^{2}=-1} . 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:
e i φ = cos φ + i sin φ , {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,}
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 φ = π {\displaystyle \varphi =\pi } in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants:
e i π + 1 = 0. {\displaystyle e^{i\pi }+1=0.}
There are n different complex numbers z satisfying z n = 1 {\displaystyle z^{n}=1} , and these are called the "nth roots of unity" and are given by the formula:
e 2 π i k / n ( k = 0 , 1 , 2 , … , n − 1 ) . {\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}
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 ( 16 9 ) 2 ≈ 3.16 {\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16} . 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.
Polygon approximation era
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion. This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (that is, 3.1408 < π < 3.1429). Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7. Around 150 AD, Greco-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. In ancient China, values for π included 3.1547 (around 1 AD), 10 {\displaystyle {\sqrt {10}}} (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556). Around 265 AD, the Cao Wei mathematician Liu Hui created a polygon-based iterative algorithm, with which he constructed a 3,072-sided polygon to approximate π as 3.1416. Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. Around 480 AD, Zu Chongzhi calculated that 3.1415926 < π < 3.1415927 {\displaystyle 3.1415926<\pi <3.1415927} and suggested the approximations π ≈ 355 113 = 3.14159292035 … {\textstyle \pi \approx {\frac {355}{113}}=3.14159292035\ldots } and π ≈ 22 7 = 3.142857142857 … {\textstyle \pi \approx {\frac {22}{7}}=3.142857142857\ldots } , which he termed the milü ('close ratio') and yuelü ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of π for the next 800 years. The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Around 1220, Fibonacci computed 3.1418 using a polygonal method devised independently of Archimedes. Italian author Dante apparently employed the value 3 + 2 10 ≈ 3.14142 {\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142} . The Persian astronomer Jamshīd al-Kāshī produced nine sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with 3 × 2 28 {\textstyle 3\times 2^{28}} sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved nine digits with a polygon of 3 × 2 17 {\textstyle 3\times 2^{17}} sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 1040 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation.
Infinite series
The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for π most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach also appeared in the Kerala school sometime in the 14th or 15th century. Around 1500, an infinite series that could be used to compute π, written in the form of Sanskrit verse, was presented in Tantrasamgraha by Nilakantha Somayaji. The series are presented without proof, but proofs are presented in the later work Yuktibhāṣā, published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes to Madhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to as Madhava series. The series for arctangent is sometimes called Gregory's series or the Gregory–Leibniz series. Madhava used infinite series to estimate π to 11 digits around 1400. In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in π calculations):
2 π = 2 2 ⋅ 2 + 2 2 ⋅ 2 + 2 + 2 2 ⋯ {\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
In 1655, John Wallis published what is now known as the Wallis product, also an infinite product:
π 2 = ( 2 1 ⋅ 2 3 ) ⋅ ( 4 3 ⋅ 4 5 ) ⋅ ( 6 5 ⋅ 6 7 ) ⋅ ( 8 7 ⋅ 8 9 ) ⋯ {\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots }
In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus, which led to the development of many infinite series for approximating π. Newton himself used an arcsine series to compute a 15-digit approximation of π in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." In 1671, James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent:
arctan z = z − z 3 3 + z 5 5 − z 7 7 + ⋯ {\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }
This series, sometimes called the Gregory–Leibniz series, equals π 4 {\textstyle {\frac {\pi }{4}}} when evaluated with z = 1 {\displaystyle z=1} . But for 1 {\displaystyle {1}} , it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit. In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for z = 1 3 {\textstyle z={\frac {1}{\sqrt {3}}}} to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:
π 4 = 4 arctan 1 5 − arctan 1 239 . {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}
Machin reached 100 digits of π with this formula. Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of π. Isaac Newton accelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):
Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including π 4 = 5 arctan 1 7 + 2 arctan 3 79 , {\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},} with which he computed 20 digits of π in one hour. In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss. Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device. In 1853, British mathematician William Shanks calculated π to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.
Rate of convergence Some infinite series for π converge faster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy. A simple infinite series for π is the Gregory–Leibniz series:
π = 4 1 − 4 3 + 4 5 − 4