Is $\sqrt2+\pi$ irrational?

Question

From this, as a layman I wonder if the same goes for $\sqrt2+\pi$? How about $\pi+\log2$?

Answer 1

Suppose the sum is rational, say $\;r\;$, but then

$$r=\sqrt2+\pi\implies r^2-2r\pi+\pi^2=2\implies \pi\;\;\text{is a root of the polynomial}$$

$$p(x)=x^2-2rx+r^2-2\in\Bbb Q[x]\;,\;\;\text{which of course is absurd}$$