An example of a nonrepeating
positional representation (necessarily representing an
irrational number) that is not normal.
sum (n = 1..infinity, b-n(n+1)/2)
That is,
.101001000100001000001000000100000001000000001...
The sequence contains only zeroes and ones, so it's obviously not normal when interpreted for any
base larger than two; and since the sequence can never contain
11, it can't be normal for base 2 either.
Because of this, we know there is a chain of subset relations:
Integers <
Rationals < Non-normals <
Reals
I don't think anything is known about the normality of irrational real
algebraic numbers; it would be nice to stick them in the middle of this chain.