On the Distribution of Numbers

Abstract

This paper examines the distribution of the mantissas of floating point numbers and shows how the arithmetic operations of a computer transform various distributions toward the limiting distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r(x) = {1 \over x \ln b} \quad (1/b \leqq x \leqq 1)$</tex> (where b is the base of the number system). The paper also gives a number of applications to hardware, software, and general computing which show that this distribution is not merely an amusing curiosity. A brief examination of the distribution of exponents is included.