# What is pi?

By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it. For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159, which is really pretty good, but if you want a better approximation you can use a computer to get it. Here’s pi to many more digits: 3.14159265358979323846.

Pi is an infinite decimal. Unlike numbers such as 3, 9.876, and 4.5, which have finitely many nonzero numbers to the right of the decimal place, pi has infinitely many numbers to the right of the decimal point.

If you write pi down in decimal form, the numbers to the right of the 0 never repeat in a pattern. Some infinite decimals do have patterns – for instance, the infinite decimal .3333333… has all 3’s to the right of the decimal point, and in the number .123456789123456789123456789… the sequence 123456789 is repeated. However, although many mathematicians have tried to find it, no repeating pattern for pi has been discovered – in fact, in 1768 Johann Lambert proved that there cannot be any such repeating pattern.

As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) pi is irrational: it cannot be written as a fraction (the ratio of two integers).

The area of a circle is pi times the square of the length of the radius, or “pi r squared”:

A = pi*r^2