  We present in this paper quantum real lines as quantum defomations of the
real numbers $\R$.Upon deforming the Heisenberg algebra $\cL$ generated by $(a,
a^\dagger)$ in terms of the Moyal $\ast$-product,we first construct q-deformed
algebras of q-differentiable functions in two cases where q is generic (not a
root of unity) and q is the N-th root of unity. We then investigate these
algebras and finally propose two quantum real lines as the base spaces of these
algebras. It is turned out that both quantum lines are discrete spaces and have
noncommutative structures.We further find, minimal length, fuzzy structure and
infinitesimal structure.
