The First Digit Phenomenon

Abstract

If asked whether certain digits in numbers collected randomly from, for example, the front pages of newspapers or from stock-market prices should occur more often than others, most people would think not. Nonetheless, in 1881, the astronomer and mathematician Simon Newcomb published a two-page article in the American Journal of Mathematics reporting an example of exactly that pheonomenon. Newcomb described his observation that books of logarithms in the library were quite dirty at the beginning and progressively cleaner throughout. From this he inferred that fellow scientists using the logarithm tables were looking up numbers starting with 1 more often than numbers starting with 2, numbers with first digit 2 more often than 3, and so on. After a short heuristic argument, Newcomb concluded that the probability that a number has a particular first significant digit (that is, first nonzero digit) d can be calculated as follows: Prob (first significant digit = d) = log10 (1 + 1/d), d = 1,2,...,9