The proof that pi is an irrational number is a little tricky. You can find it easily with a Google search:

In fact, we have this continued fraction:

$\LARGE\pi=3+\frac{1^2}{6+\frac{3^2}{6+\frac{5^2}{6+\frac{7^2}{6+\frac{9^2}{6+...}}}}}$

$\pi$ is more than just an irrational number, it is a transcendental number.

An immediate consequence of Lindemann's proof of the transcendence of $\pi$ is that it proves the impossibility of the geometric problem of antiquity known as squaring the circle i.e. constructing a square with the same area as a circle with ruler and compass only.

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