Why can we write almost any nonzero 2-adic integer in the same form?
The pattern $x = (\alpha01^a10^b)_2$ is very easy to match. For example, the number $1=0^\infty1$ matches the pattern with $\alpha=0^\infty, a=b=0$. Try matching some other numbers yourself. The only thing this pattern asks for, is for the string "$01$" to appear somewhere in the sequence of bits. The only numbers to fail this condition are $0=0^\infty$ and $-2^b=1^\infty0^b$.