If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension?

The dimension of $\mathbb{R}$ as a vector space over $\mathbb{Q}$ is equal to the cardinality of $\mathbb{R}$.

In general, the dimension and cardinality of any vector space $\mathbf{V}$ and the cardinality of its scalar field $\mathbf{K}$ will obey the following equation:

$$ |\mathbf{V}| = \begin{cases} |\mathbf{K}|^{\dim(\mathbf{V})} & \text{$\dim(\mathbf{V})$ is finite} \\ \max(|\mathbf{K}|, \dim(\mathbf{V})) & \text{$\dim(\mathbf{V})$ is infinite} \end{cases}$$

In the case of $\mathbb{R}$ over $\mathbb{Q}$, if the dimension of $\mathbb{R}$ were somehow finite then this equation would imply that $|\mathbb{R}|$ is equal to some finite power of $|\mathbb{Q}|$, which is false (every finite power of a countable set is countable). Since $|\mathbb{R}|$ is greater than $|\mathbb{Q}|$, the second branch of the equation implies that the dimension of $\mathbb{R}$ is equal to $|\mathbb{R}|$.

(For a quickie proof of the infinite-dimensional case, see https://math.stackexchange.com/a/194287.)

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