Wolfram Mathworld explains a Hamel basis as "a set of real numbers $\left\{U_\alpha\right\}$ such that every real number $\beta$ has a unique representation of the form:
$$ \beta = \sum\limits_{i=1}^{n}r_i U_{\alpha_i} $$
where $r_i$ is rational and $n$ depends on $\beta$."
Am I correct to interpret this as meaning that every real number is a unique finite linear combination of real numbers from a proper subset of $\mathbb{R}$ with coefficients in $\mathbb{Q}$? This would be quite a fascinating result to me.