= Solution
Let $K$ have $r$ real embeddings and $s$ conjugate pairs of complex embeddings. The <Dirichlet unit theorem> states that
$$
\mathcal O_K^\times\cong\mu(K)\times\mathbb Z^{r+s-1},
$$
where $\mu(K)$ is the finite cyclic group of roots of unity in $K$. Thus the unit rank is
$$
m=r+s-1.
$$
Equivalently, there are units $u_1,\ldots,u_m$ such that every unit has a unique expression
$$
\boxed{u=\zeta u_1^{n_1}\cdots u_m^{n_m},
\qquad
\zeta\in\mu(K),\quad n_i\in\mathbb Z.}
$$
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