Number of elements in the cyclic group

Question

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)?

Can anyone please help?

Thanks in advance!

Answer 1

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 $(\Bbb Z,+)$.