Normal Numbers Are Normal

A number is normal in base b if every sequence of k symbols in the letters 0, 1, . . . , b− 1 occurs in the base-b expansion of the given number with the expected frequency b−k. From an informal point of view, we can think of numbers normal in base 2 as those produced by flipping a fair coin, recording 1 for heads and 0 for tails. Normal numbers are those which are normal in every base. In this expository article, we recall Borel’s result that almost all numbers are normal. Despite the abundance of such numbers, it is exceedingly difficult to find specific exemplars. While it is known that the Champernowne number 0.123456789101112131415 · · · is normal in base 10, it is (for example) unknown whether √ 2 is normal in any base. We sketch a bit of what is known and what is not known of this peculiar class of numbers, and we discuss connections with areas such as computability theory.