The fact that the quaternions “work” (are a finite-dimensional real vector space with an associative real-bilinear multiplication, and a division operation) is quite special. In fact there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions.

> there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions.

Does a proof exist that these are the only three of such objects?

Yes, and the proof only uses linear algebra: https://en.wikipedia.org/wiki/Frobenius_theorem_(real_divisi...

Thanks. Great result, surprising for 1877.

In mathematics often proofs lead to understanding why something works (on the other hand, many proofs aren’t written in a way that make the main ideas clear or easy to tease out from details)