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The unit rank of a number field $K$ is the rank of the free abelian part of the unit group of its ring of integers. If $K$ has $r_1$ real embeddings and $r_2$ conjugate pairs of complex embeddings, Dirichlet's unit theorem giveswhere $\mu (K)$ is the finite group of roots of unity in $K$. Hence the unit rank is $r_1+r_2-1$.^[ref]

The rational numbers and imaginary quadratic fields have rank zero; real quadratic fields have rank one. In $\mathbb{Q}(\sqrt{2})$, the unit $1+\sqrt{2}$ has norm $-1$ and its powers produce infinitely many units. The logarithms of the absolute embedding values of a unit sum to zero when complex pairs are counted twice. This places their logarithmic images in a hyperplane of dimension $r_1+r_2-1$, explaining the geometric meaning of the rank.