On the Normality of Numbers

A number is normal in a base if, in its expansion to that base, all possible digit strings of any given length are equally frequent. While it is generally believed that many familiar irrational constants are normal, normality has only been proven for numbers expressly invented for the purpose of proving their normality. In this study we review some of the main results to date. We then define a new normality criterion, strong normality, to exclude certain normal but clearly non-random artificial numbers. We show that strongly normal numbers are normal but that Champernowne's number, the best-known example of a normal number, fails to be strongly normal. We also re-frame the question of normality as a question about the frequency of residue classes of an increasing sequence of integers modulo some fixed integer. This leads to the beginning of a detailed examination of the digits of irrational square roots.