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.