  It turns out that some empirical facts in Big Data are the effects of
properties of large numbers. Zipf's law 'noise' is an example of such an
artefact. We expose several properties of the power law distributions and of
similar distribution that occur when the population is finite and the rank and
counts of elements in the population are natural numbers. We are particularly
concerned with the low-rank end of the graph of the law, the potential of noise
in the law, and with the approximation of the number of types of objects at
various ranks. Approximations instead of exact solutions are the center of
attention. Consequences in the interpretation of Zipf's law are discussed.
