# How to find a basis for the vector space of real numbers over the field Q?

So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers that are not algebraic. On the other hand we know from the axiom of choice that any vector space has a basis, so is there a way to find a basis for this interesting one?

I hope my question isn't wrong.

I think it might work. For example, if $[r]_{\alpha}$ is the collection of equivilence classes, and if $z \in [0,1]$ and $z \in [r]$ for some $r$ then $r-z = q \in \mathbb{Q}$. So that $z = 1r + q1$. So if we require that 1 be one of the numbers from the equivilency classes, then this set certainly spans $[0,1]$. And by taking $q$ to be an integer + q it seems like this set will span the whole real line. Now, are they linearly independent?