Unique factorization domain: existence of a complex number

Question

Let $w = (-1 + i \sqrt{3})/2$ and $\mathbb Z[w] = \{ m + nw \mid m, n \in \mathbb Z\}$.

Why is there for every $z \in \mathbb C$ a $q \in \mathbb Z[w]$ such that $|z-q| < 1$?

Answer 1

This is because $\mathbf Z[ω]$ (seen in the Argand-Cauchy plane) is a lattice in $\mathbf R^2$, and its fundamental region is a rhombus with sides equal to $1$. So any point in the rhombus is inside one of the unit circles centred at one of the vertices of the rhombus.