The standard proof is by contradiction.

If they were both rational, then

If they were both rational, then

would also be rational, which implies that (π−e) is an algebraic number (i.e., the root of some nonzero polynomial with rational coefficients).

Since π+e is rational and thus algebraic by hypothesis, it follows that (π−e)+(π+e)=2π is algebraic (being the sum of two algebraic numbers) and hence so is π.

Similarly, we can infer that e is algebraic.

However, both are known to be transcendental — contradiction.

Since π+e is rational and thus algebraic by hypothesis, it follows that (π−e)+(π+e)=2π is algebraic (being the sum of two algebraic numbers) and hence so is π.

Similarly, we can infer that e is algebraic.

However, both are known to be transcendental — contradiction.