The Uneven Distribution of Numbers in Nature

Suppose you look at today's stock prices and bet on the value of the first digit. One could guess that a fair bet should correspond to the frequency of $1/9 = 11.11%$ for each digit from 1 to 9. This is by no means the case, and one can easily observe a strong prevalence of the small values over the large ones. The first three integers 1,2 and 3 alone have globally a frequency of 60% while the other six values 4, 5, 6, 7, 8 and 9 appear only in 40% of the cases. This situation is actually much more general than the stock market and it occurs in a variety of number catalogs related to natural phenomena. The first observation of this property traces back to S. Newcomb in 1881 but a more precise account was given by F. Benford in 1938. In this note we illustrate these observations with the enlightening specific example of the stock market. We also identify the general mechanism for the origin of this uneven distribution in the multiplicative nature of fluctuations in economics and in many natural phenomena. This provides a natural explanation for the ubiquitous presence of the Benford's law in many different phenomena with the common element that their fluctuations refer to a fraction of their values. This brings us close to the problem of the spontaneous origin of scale invariant properties in various phenomena which is a debated question at the frontier of different fields.