The standard proof is by contradiction.
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.
Parametric study of cantilever plates exposed to supersonic
and hypersonic flows.