  We investigate, using Monte Carlo simulations, the phase diagram of a system
of hard rectangles of size $m\times mk$ on a square lattice when the aspect
ratio $k$ is a non-integer. The existence of a disordered isotropic phase, a
nematic with only orientational order, a columnar phase with orientational and
partial translational order, and a high density phase with no orientational
order is shown. The high density phase is a solid-like sublattice phase only if
the length and width of the rectangles are not mutually prime, else, it is an
isotropic phase. The minimum value of $k$ beyond which the nematic and columnar
phases exist are determined for $m=2$ and $3$. The nature of the transitions
between different phases is determined, and the critical exponents are
numerically obtained for the continuous transitions.
