  Efforts to understand the microscopic origin of superconductivity in the
cuprates are dependent on knowledge of the normal state. The Hall number in the
low temperature, high field limit $n_{\rm H}(0)$ has a particular significance
because within conventional transport theory it is simply related to the number
of charge carriers, and so its evolution with doping gives crucial information
about the nature of the charge transport. Here we report a study of the high
field Hall coefficient of the single-layer cuprates
Tl$_2$Ba$_2$CuO$_{6+\delta}$ (Tl2201) and (Pb/La) doped
Bi$_2$Sr$_2$CuO$_{6+\delta}$ (Bi2201) which shows how $n_{\rm H}(0)$ evolves in
the overdoped, so-called strange metal, regime of cuprates. We find that
$n_{\rm H}(0)$ increases smoothly from $p$ to $1+p$, where $p$ is the number of
holes doped into the parent insulating state, over a wide range of doping. The
evolution of $n_{\rm H}$ correlates with the emergence of the anomalous
linear-in-$T$ term in the low-$T$ in-plane resistivity. The results could
suggest that quasiparticle decoherence extends to dopings well beyond the
pseudogap regime.
