Pi ( $π$ )

Pi is the ratio of a Euclidean circle's circumference to its diameter.

Table of contents

Chudnovsky algorithm

Chudnovsky series

Wallis product ( $π /2 = \prod_{j \geq 1} \frac{4 j ^{2}}{4 j ^{2} - 1}$ )

The Wallis product is

\frac{π}{2} = \prod_{j = 1}^{\infty} \frac{( 2 j ) ^{2}}{( 2 j - 1 ) ( 2 j + 1 )} .

Its finite product is

W_{n} = 2^{4 n} (n!)^{4} / [(2 n + 1) ((2 n)!)^{2}]

. The Wallis integrals give

2 W_{n} / π = I_{2 n + 1} / I_{2 n} \to 1

, proving both convergence of the product and its value. The strict integral bounds also give

πn / (2 n + 1) < W_{n} < π /2

.

Proof that π is irrational

Assuming

π = a / b

and integrating a suitably normalized polynomial times

sin x

produces a positive integer that tends to zero as the polynomial degree grows. This contradiction proves that

π

is an irrational number.

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Pi by

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Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning that it cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal representation. The value of Pi is approximately 3.14159, but it extends infinitely beyond the decimal point. Pi is widely used in mathematics, physics, engineering, and various fields that involve circular or cyclical phenomena.

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Pi by

Ciro Santilli  40 Updated 2025-07-16

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