Why is it so hard to prove a number is transcendental?
Solution 1:
For example, $\pi$ and $1-\pi$ are transcendental, but $\pi+(1-\pi)=1$ is not.
Solution 2:
It is mainly because transcedental numbers behave so weird. For example, would you think that a transcedental number raised to an irrational is an integer? Well, it is possible:
$$(2^{\sqrt{2}})^{\sqrt{2}} = 2^{(\sqrt{2})^2}=4$$