Is notation of a number in base-(-2) numeral system unambiguous?

Question

I have a problem with proving this thesis:

Notation of a number in base-(-2) numeral system is unambiguous.

I think I need to use mathematical induction, but I don't know how.

Answer 1

You want to show that if $$\sum_{i=0}^\infty a_i(-2)^i = \sum_{i=0}^\infty b_i(-2)^i$$ are equal, for $a_i, b_i \in {0,1}$, then the coefficients are equal: $a_i=b_i$.

Suppose otherwise, for contradiction, and let $N$ be the highest index for which the two sequences differ (this requires us to assume that both sequences are zero for sufficiently high index, i.e. that the numbers have finitely many non-zero base -2 digits.)

Then $$0 = \sum_{i=0}^N (a_i-b_i)(-2)^i = (a_N-b_N)(-2)^N+\sum_{i=0}^{N-1} (a_i-b_i)(-2)^i.$$

Now $(a_N-b_N)(-2)^N$ is equal to either $2^N$ or $-2^N$ (it can't be zero, by assumption); assume without loss of generality the latter. Then $$0 = -2^N + \sum_{i=0}^{N-1} (a_i-b_i)(-2)^i \leq -2^N + \sum_{i=0}^{N-1} |a_i-b_i|2^i < -2^N + 2^N -1 = -1,$$ a contradiction. So representations in base -2 are unique.