A bijection between two different basis for different vector spaces

Question

It is clear that if {$r_j$}$_{j \in J}$ is a $\mathbb{Q}$-basis for $\mathbb{R}$, then {$r_j$}$_{j \in J}$ $\cup$ {$ir_j$}$_{j \in J}$ is a $\mathbb{Q}$-basis for $\mathbb{C}$. Also, assuming these two basis have the same cardinality, I am able to show that $\mathbb{R}$ and $\mathbb{C}$ are isomorphic as $\mathbb{Q}$ vector spaces. However I am not able to prove that there is a bijection between the two basis, that is the part I need help with. Thanks!

Answer 1

In terms of cardinal arithmetic, if $\kappa$ is an infinite cardinal then $\kappa + \kappa = 2 \kappa = \kappa$. Here $\kappa + \kappa$ is the cardinality of the disjoint union of two sets of cardinality $\kappa$. This fact alone proves that the two basis are of equal size. It can also be show that since $\mathbb{Q}$ is countable and thus of smaller cardinalty than $\mathbb{R}, \mathbb{C}$, the basis must also have cardinality of the continuum.