Multiple origins of the Newcomb-Benford law: rational numbers, exponential growth and random fragmentation

The Newcomb-Benford law states that, in data drawn randomly from many different sources, the probability that the first significant digit is n is given by log(1 1/n). In a previous paper, it was shown that there are at least two basic mechanisms for this phenomenon, depending on the origin of the data. In the case of physical quantities measured with arbitrarily defined units, it was shown to be a consequence of the properties of the rational numbers, whereas for data sets consisting of natural numbers, such as population data, it follows from the assumption of exponential growth. It was also shown that, contrary to what has been maintained in the literature, the requirement of scale invariance alone is not sufficient to account for the law. The present paper expands on the previous paper, and it is shown that the finite set of rational numbers to which all measurements belong automatically satisfies the requirement of scale invariance. Further, a third mechanism, termed random fragmentation , is proposed for natural number data which are not subject to exponential growth. In this case, however, the Newcomb-Benford is only approximately reproduced, and only under a certain range of initial conditions.