Pi

Math

Pi is an irrational number and a transcendental number. It is important in mathematics, and helps in geometry, where circles are used in.

It is equal to 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 5923 (to 64 decimal places).

In geometry


An SI unit called radians define pi radians to be 180 degrees.

So, it is also used in trigonometry.

A circle defines pi to be the circumference (C) of the circle divided by the diameter (d) of the circle. As all circles are similar, the ratio is always pi.

A circle has an infinite number of sides, so pi can be defined as \(\lim_{n\rightarrow\infty}n\tan{\frac{\pi}{n}}\), where n is the number of sides of a polygon.

Uses
In other fields, it is also important.

Euler's identity and formula
Euler's formula uses radians in here: usually \(e^{ix}=\cos x+i\sin x=\operatorname{cis}x\), so a value of pi will form Euler's identity, equal to -1.

Tau (\(\tau\))
It is more accepted than tau, where \(\tau=2\pi\).

Continued fraction expansion
It is [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, ...], or \[\pi = 3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\ddots}}}}\]

The continued fraction expansion does not repeat, unlike another related transcendental number, e.

Approximations
$3⁄1$, $22⁄7$, $333⁄106$, $355⁄113$, $103993⁄33102$, ...