Transcendental Basis

Question

Can you say that because $\pi$ is transcendental, that a basis of $\{\pi, \pi^2, \pi^3, \dots\}$ in the rational numbers $\mathbb{Q}$ spans the entire real numbers? It seems likely, although I can't think of a proof of the fact.

Answer 1

No, and in fact any basis for $\mathbb{R}$ over $\mathbb{Q}$ must be uncountable. One can see this in the following way: if $B$ is a countable set of real numbers, then the span of $B$, the set of all finite linear combinations (with rational coefficients) of elements of $B$, is also a countable set, but $\mathbb{R}$ is uncountable.