I know almost nothing about transcendental numbers, I know the definition of them and maybe few results about them and that is all.
But the question in the title somehow naturally arises when thinking about transcendental numbers.
I think that it is okay to state it once more in the body of the question and not only in the title so here is the question again:
Suppose that $\alpha$ is some transcendental number and that $\beta$ is algebraic number. Is the sum $\alpha + \beta$ always transcendental?