Does $\Bbb Z_2^\times$ contain non-repeating infinitely long binary strings?

Question

Does $\Bbb Z_2^\times$ contain non-repeating infinitely long binary strings?

I appreciate we have infinitely long repeating binary strings such as $\overline{01}_2=-\frac{1}{3}$

And of course we have finitely long strings such as $01_2=1$

And ultimately-repeating strings in $\Bbb Z_2^\times$ such as $\frac23=\overline{01}10_2$ can be represented as sums and differences of these.

But does $\Bbb Z_2^{\times}$ also contain non-repeating infinite binary strings? I've seen reference to square-roots and such things which it seems to stand to reason might be represented by non-repeating binary strings.

It seems obvious the answer's the same for $\Bbb{Z}_2$ and $\Bbb Q_2$

Answer 1

The expansion is repeating if and only if it's rational. $\sqrt{17}$ is a 2-adic integer, so it's non-repeating.