Why do Quaternions and octonions exist?

Question

I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are closed under addition, multiplication and exponentiation.

What are the quaternions and octonions (I remember seeing $j$ and $k$) and other hypercomplex systems (as they are called), and why did we create them?

Also, I remember reading that the octonions are the largest of these hypercomplex systems (meaning that any number in a hypercomplex system is also a number in the system of the octonions).

Thank you very much in advance.

Answer 1

The place to look is John Baez's beautifully written article on The Octonions. The introduction is wonderfully entertaining and the relevant section you want to focus on is the Cayley-Dickson construction.