  We revisit the relationship between three classical measures of particle
number, namely the chemical doping $x$, the Hall number $x_{hall}$ and the
particle number inferred from the optical sum rule $x_{opt}$. We study the
$t$-$t'$-$J$ model of correlations on a square lattice, as a minimal model for
High $T_c$ systems, using numerical methods to evaluate the low temperature
Kubo conductivites. These measures disagree significantly in this type of
system, owing to Mott Hubbard correlations. The Hall constant has a complex
behavior with several changes of sign as a function of filling $x$, depending
upon the model parameters. Thus $x_{hall}$ depends sensitively on $t'$ and $J$,
due to a kind of quantum interference.
