# Number of elements in the cyclic group

I recently came across this question:

Find the number of elements in the following cyclic group:

The cyclic sub-group of $$C^{*}$$ generated by 1+i

The solution says the answer is: O(Z), where Z is the set of all natural numbers. It further says this is so because $$|1+i|=\sqrt{2} >1$$.

I couldn't get it intuitively. Why does the order have to be O(Z)?

If $$z=1+i$$, then $$|z|=\sqrt2>1$$. Then $$|z^2|>|z|$$, $$|z^3|>|z^2|$$, and so on. So, $$C^*$$ is an infinite cyclic group, and therefore it is isomorphic to $$(\Bbb Z,+)$$.