# Pi

π&pi;\pi3.14ratio of a circle's circumference to its diameter (pi)3.1416circumference-to-diameter ratio3.14159
The number is a mathematical constant.wikipedia
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### Mathematical constant

constantconstantsmathematical constants
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 Approximating approximation
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

irrationalirrational numbersirrationality
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.

### Squaring the circle

square the circlequadrature of the circlesquaring of 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.

### Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
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.

### Transcendental number

transcendentaltranscendencetranscendental irrational
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients.
The best-known transcendental numbers are e''.

### Piphilology

digit span memorizationmemorize the digitsmemorize the value of
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|]].

### Real number

realrealsreal-valued
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...).

### Normal number

normalabsolutely normal numberequidistributed
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 √2, e'' are normal, but a proof remains elusive.

### Circle

circularcircles360 degrees
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 256⁄81 (3.16049...) as an approximate value of [[Pi|]].

### Rational number

rationalrational numbersrationals
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients.
]], [[Pi|]], [[E (mathematical constant)|

### Proof that π is irrational

Proof that is irrational is irrationalbelow
There are several proofs that is irrational; they generally require calculus and rely on the reductio ad absurdum technique.
In the 1760's, 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.

### History of mathematics

historian of mathematicsmathematicshistory
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, 310⁄71 < π < 310⁄70.

### Repeating decimal

recurring decimalrepeatrepeating fraction
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).
Examples of such irrational numbers are the square root of 2 and [[pi|]].

### Statistical randomness

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.

### Circumference

circumferentialgirthcircumferential line
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.
This constant, pi, is represented by the Greek letter [[Pi (letter)|]].

### Pi (letter)

πpiϖ
It has been represented by the Greek letter "[[Pi (letter)|]]" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.
The mathematical real transcendental (and thus irrational) constant π = 3.14159..., the ratio of a circle's circumference to its diameter in Euclidean geometry. The letter "π" is the first letter of the Greek words "περιφέρεια" 'periphery' and "περίμετρος" 'perimeter', i.e. the circumference.

### Chinese mathematics

Chinese mathematicianmathematicsChinese mathematical
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.
However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used.

### Euler's identity

Euler's identity, e^{i\pi}=-1
= in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:
: is pi, the ratio of the circumference of a circle to its diameter.

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|]].

### Algebraic number

algebraicalgebraic numbersAlgebraic form
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)|]].

### Geometry

geometricgeometricalgeometries
Because its most elementary definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres.
Archimedes (c. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.

### Six nines in pi

999999sequence of six consecutive 9s
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 the number pi, starting at the 762nd decimal place.

### Complex number

complexreal partimaginary part
, 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.

### Ratio

ratiosproportionratio analysis
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 earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal to the length of a side of a square, which is the square root of 2, formally Another example is the ratio of a circle's circumference to its diameter, which is called algebraically irrational number, but a transcendental irrational.