Gaussian integer cardinality

Question

I would like to what cardinal number the group of Gaussian integer $\Bbb Z[i]$ when $i^2=-1$, is similar to? Is it $\Bbb Z$?

Answer 1

We know that the cartesian product of numerable sets is numerable. In this case we can visualize the elements of your group as the result of the cartesian product Z x Z. So we have a numerable set. As you said it is the same cardinality of the integers.