TITLE: Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
QUESTION [163 upvotes]: It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?

REPLY [5 votes]: As $\pi$ is trascendent over $\mathbb{Q}$. Then the set $\{1, \pi, \pi^{2},\cdots\}$ is linearly independent.