  We introduce the integrality number of an integer program (IP) in inequality
form. Roughly speaking, the integrality number is the smallest number of
integer constraints needed to solve an IP via a mixed integer (MIP) relaxation.
One notable property of this number is its invariance under unimodular
transformations of the constraint matrix. Considering the largest minor
$\Delta$ of the constraint matrix, our analysis allows us to make statements of
the following form: there exist numbers $\tau(\Delta)$ and $\kappa(\Delta)$
such that an IP with $n\geq \tau(\Delta)$ many variables and $n +
\kappa(\Delta)\cdot \sqrt{n}$ many inequality constraints can be solved via a
MIP relaxation with fewer than $n$ integer constraints. From our results it
follows that IPs defined by only $n$ constraints can be solved via a MIP
relaxation with $O(\sqrt{\Delta})$ many integer constraints.
