Pi is Transcendental

Theorem
$\pi$ (pi) is transcendental.

Proof
Proof by Contradiction:

$\pi$ is not transcendental.

Hence by definition, $\pi$ is algebraic.

We have that $i$ is a root of $z^2 + 1 = 0$.

Hence $i$ is algebraic.

It follows from Algebraic Numbers are Closed under Multiplication that $i \pi$ is also algebraic.

From the Weaker Hermite-Lindemann-Weierstrass Theorem, $e^{i \pi}$ is transcendental.

However, from Euler's Identity:
 * $e^{i \pi} = -1$

which is the root of $z + 1 = 0$, and so is algebraic.

This contradicts the conclusion that $e^{i \pi}$ is transcendental.

Hence by Proof by Contradiction it must follow that $\pi$ is transcendental.