Countability of Gaussian Integers

Question

I am attempting to show that Gaussian Integers are countable.

Is it valid to map $a + bi$ to an ordered pair $(a, b)$ and then map this to the set of rationals $a/b$ ?

I am unsure if this works since $a/b$ is not defined at $b = 0$ and am unsure of a different way to go about this. Any hints welcome, thanks!

Answer 1

We have $\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\}$ and thus a bijection to pairs $(a,b)\in \mathbb{Z}\times \mathbb{Z}$. Since the product of two countable sets is countable, this is countable.