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**

### Mathematical constant

**mathematical constantconstantconstants**

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 π

**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 number

**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 π

**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

### Piphilology

**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

**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 circle

**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.

### Real number

**real numberrealreals**

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...).

### Transcendental number

**transcendental numbertranscendentaltranscendence**

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 π is irrational

**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 number

**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 randomness

**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 number

**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|]].

### Circle

**circlecircularcircles**

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 mathematics

**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 number

**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 Kanada

**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|]].

### Euler's identity

Setting = in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:

### Wallis product

**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

### Six nines in pi

**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 algorithm

**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|]].

### Ferdinand von Lindemann

**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.

### Viète's formula

**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 number

**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.

### Milü

**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 .