The number is a mathematical constant.
piππ ratio of a circle's circumference to its diameter3.14 (pi)the ratio of the circumference of a circle to its diametercircumference-to-diameter ratio3.1416
The number is a mathematical constant.
Constants arise in many areas of mathematics, with constants such as [[e (mathematical constant)|]] and [[pi|]] occurring in such diverse contexts as geometry, number theory, and calculus.
Approximations of approximationapproximation of π
Still, fractions such as 22/7 and other rational numbers are commonly used to approximate.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes).
irrational numberirrationalirrational numbers
Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern).
Among irrational numbers are the ratio [[Pi|]] of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.
Leibniz formula for Madhava–Leibniz seriesGregory–Leibniz series
The historically first exact formula for, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics.
In mathematics, the Leibniz formula for [[Pi|]], named after Gottfried Leibniz, states that
piphilologymemorize the value of piem
Attempts to memorize the value of with increasing precision have led to records of over 70,000 digits.
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant [[pi|]].
Archimedes of SyracuseArchimedesArchimedean
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques.
Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers.
squaring the circlequadrature of the circlesquare the circle
This transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.
In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry.
Included within the irrationals are the transcendental numbers, such as [[pi|]] (3.14159265...).
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients. In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler.
The best-known transcendental numbers are e''.
Proof that is irrationalproof that π is irrational is irrational
There are several proofs that is irrational; they generally require calculus and rely on the reductio ad absurdum technique.
In the 18th century, Johann Heinrich Lambert proved that the number irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer.
normal numbernormalabsolutely normal number
In particular, the digit sequence of is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. 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.
It is widely believed that the (computable) numbers e'' are normal, but a proof remains elusive.
statistical randomnessrandomrandom 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.
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness.
rational numberrationalrational numbers
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients.
Irrational numbers include [[square root of 2|]], [[Pi|]], [[E (mathematical constant)|]], and [[Golden ratio|]].
Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.
The result corresponds to (3.16049...) as an approximate value of [[Pi|]].
history of mathematicshistorian of mathematicsmathematics
Ancient civilizations required fairly accurate computed values to approximate for practical reasons, including the Egyptians and Babylonians.
He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 3 < π < 3.
algebraic numberalgebraicalgebraic numbers
Because is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational.
They include [[Pi|]] and [[e (mathematical constant)|]].
Yasumasa KanadaKanada, Yasumasa
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.
Yasumasa Kanada is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of [[pi|]].
Setting = in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:
Wallissecond infinite sequence found in EuropeWallis' product
The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product:
In mathematics, Wallis' product for [[Pi|]], written down in 1655 by John Wallis, states that
999999sequence of six consecutive 9ssix nines in pi
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.
A sequence of six 9s occurs in the decimal representation of π, starting at the 762nd decimal place.
Chudnovsky's seriesChudnovsky brothersfamily of formulas
Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.
The Chudnovsky algorithm is a fast method for calculating the digits of [[pi|]].
LindemannC. L. Ferdinand LindemannLindemann, Carl Louis Ferdinandvon
In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler.
Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that [[pi|]] (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.
first infinite sequence discovered in EuropeViète productViète's infinite product
The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in calculations) found by French mathematician François Viète in 1593:
In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant [[pi|]]:
complex numbercomplexreal part
In a similar spirit, can be defined instead using properties of the complex exponential, of a complex variable.
As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.
approximated pi355/113 355 ⁄ 113
The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon.
The name Milü ("close ratio"), also known as Zulü (Zu's ratio), is given to an approximation to astronomer, Zǔ Chōngzhī .