  We establish a link between the new theory of $q$-deformed rational numbers
and the classical Burau representation of the braid group $\mathrm{B}_3$. We
apply this link to the open problem of classification of faithful complex
specializations of this representation. As a result we provide an answer to
this problem in terms of the singular set of the $q$-rationals and prove the
faithfulness of the Burau representation specialized at complex $t\in
\mathbb{C}^*$ outside the annulus $3-2\sqrt2 \leq |t| \leq 3+2\sqrt2.$
