  We consider $q$-binomial coefficients built from the $q$-rational and
$q$-real numbers defined by Morier-Genoud and Ovsienko in terms of continued
fractions. We establish versions of both the $q$-Pascal identity and the
$q$-binomial theorem in this setting. These results are then used to find more
identities satisfied by the $q$-analogues of Morier-Genoud and Ovsienko,
including a Chu--Vandermonde identity and $q$-Gamma function identities.
