consider R as a vector space over Q then what is the dimension and how do we prove it
marked as duplicate by Akhil Mathew Mar 21 '11 at 14:10
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The dimension is infinite; it suffices to show that the powers of, say, $\pi$ are linearly independent (a nontrivial finite linear combinations of powers of $\pi$ that equals 0 would show $\pi$ to be an algebraic number, which it is not).