# Another Fun Arithmetic Fact: The Go Ghoti Theorem

Theorem. For every natural number $n$,

$(-1)^n = e$

Proof. $n$ is an ordinal, and hence equals $\{0, 1, \dots, n-1\}$. Since $n$ is an object in the category Set, it also represents the identity morphism on that object. Of course, $(-1)^{\sigma}$ for a permutation $\sigma$ is the sign of the permutation. Since the identity permutation is even, this equals the identity element $e$ of the group $\mathbb {R}^{\times}$.