Normal Numbers without Measure Theory
Rodney Nillsen · American Mathematical Monthly · 2000
Any number can be expanded to the base 10, leading to a sequence of digits between 0 and 9 corresponding to the number.Also, any number can be expanded to the base 2, leading to a sequence of digits, each one being either 0 or 1, corresponding to the number.It is result due to Émile Borel in 1904 that "almost all" numbers have the property that, when expanded to the base 2, each of the digits 0 and 1 appears with an asymptotic frequency of 1/2.That is, if we regard the sequence of digits in the expansion to the base 2 as a sequence of 'heads' and 'tails' resulting from a coin-tossing experiment, then, in the language of probability theory, the probability of getting heads (that is a 0) is 1/2, and the probability of getting tails (that is a 1) is also 1/2.Numbers with this property are called "simply normal numbers" to the base 2. Traditionally, the proof of Borel's Theorem relies on a knowledge of measure theory, which generally lies outside the undergraduate curriculum.Here, a proof of Borel's Theorem is presented which requires only an introductory knowledge of sequences and series, and a knowledge of how to integrate step functions on an interval.This makes it possible to discuss Borel's theorem at the level of a first or second year course in mathematical analysis.