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 Hurwitzs 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.
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