Normal numbers

Imagine if we sample numbers from the unit interval and count the occurrences of each digit in every number’s infinite decimal representation. For instance, in the number , every non-zero digit appears exactly once, but appears infinitely often (trailing zeros). On the other hand, for the number , the digit occurrences (after the “dot”) is equal for every digit. As the number of samples increases, one may expect that the number of occurrences for each digit in to be close to one another. In fact, the Normal Number Theorem, first formulated by Borel, not only formalizes this intuition but goes a step further: for almost every number in , each digit appears […]

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