# Why do Quaternions and octonions exist?

Ok so 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. So my question is first of all: what are the quaternions and octonions (I remember seeing thinks like $j$ and $k$) and other Hypercomplex systems (as they are called). And second of all why did we create them. Also, I remember reading that the octonions are the largest of these hyperecomplex systems (meaning that any number in a hypercomplex system is also a number in the system of the octonions). Than you very much in advance.