It has admittedly been quite a while since my last post over a year ago. I thought I would restart the posts by revisiting one of the first topics I discussed on the website: quaternions. My previous post, upon review, seems to be quite uninformative on what the nature and use of them are which I will attempt to show in this post.

## Formalism

Quaternions are a generalization of complex numbers () or hypercomplex numbers and they are denoted with . Below I write both in their general form.

where

Now, there are 3 “imaginary” components and they are defined by that relation at the bottom. This is super interesting! What does this even mean though? A real number with some sort of 3-dimensional imaginary component?