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}.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s