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.
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