Pi is Transcendental

Hermite-Lindemann-Weierstrass Theorem Let  be algebraic numbers (possibly complex) that are linearly independent over the rational numbers . Then:  are algebraically independent.

Weaker Hermite-Lindemann-Weierstrass Theorem Let  be a non-zero algebraic number (possibly complex). Then:  is transcendental

Suppose  is not transcendental.

Hence by definition,  is algebraic.

Let  be the root of a non-zero polynomial with rational coefficients, namely .

Then,  is also a non-zero polynomial with rational coefficients such that: 

Hence,  is also algebraic.

From the Weaker Hermite-Lindemann-Weierstrass Theorem ,  is transcendental.

However, from Euler's Identity:

which is the root of  and so is algebraic.

This contradicts the conclusion that  is transcendental.

Hence by Proof by Contradiction it must follow that  is transcendental.