Normal Schmormal: My occasionally helpful guide to parenting kids with special needs (Down syndrome, autism, ADHD, neurodivergence)

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Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point). It has also been conjectured that every irrational algebraic number is absolutely normal (which would imply that √ 2 is normal), and no counterexamples are known in any base. BaileyandCrandall( 2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers. It has been an elusive goal to prove the normality of numbers that are not artificially constructed. The set of non-normal numbers, despite being "large" in the sense of being uncountable, is also a null set (as its Lebesgue measure as a subset of the real numbers is zero, so it essentially takes up no space within the real numbers).

For example, in a normal binary sequence (a sequence over the alphabet { 0, 1}), 0 and 1 each occur with frequency 1⁄ 2; 00, 01, 10, and 11 each occur with frequency 1⁄ 4; 000, 001, 010, 011, 100, 101, 110, and 111 each occur with frequency 1⁄ 8; etc.If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length.

For bases r and s with log r / log s irrational, there are uncountably many numbers normal in each base but not the other.

We say that x is simply normal in base b if the sequence S x, b is simply normal [5] and that x is normal in base b if the sequence S x, b is normal. In mathematics, a real number is said to be simply normal in an integer base b [1] if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density1/ b. For bases r and s with log r / log s rational (so that r = b m and s = b n) every number normal in base r is normal in base s.