Infinite non-periodic binary fraction

Question

I have an infinite non-periodic binary fraction. For example:

$frac_1=0.101111011100110010001001010010000001001...$

Is it always true that $1-frac_1$ = non-periodic binary fraction?

Answer 1

Well, yes. This is actually quite obvious.

If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).

Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.

In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.