  We study analytic properties of "$q$-deformed real numbers", a notion
recently introduced by two of us. A $q$-deformed positive real number is a
power series with integer coefficients in one formal variable~$q$. We study the
radius of convergence of these power series assuming that $q \in \C.$ Our main
conjecture, which can be viewed as a $q$-analogue of Hurwitz's Irrational
Number Theorem, provides a lower bound for these radii, given by the radius of
convergence of the $q$-deformed golden ratio. The conjecture is proved in
several particular cases and confirmed by a number of computer experiments. For
an interesting sequence of "Pell polynomials", we obtain stronger bounds.
