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}.

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