8
$\begingroup$

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.

  • 1
    You could have a look at this: http://math.stackexchange.com/questions/529/why-are-the-only-division-algebras-over-the-real-numbers-the-real-numbers-the-c?rq=1. Also, sedenions arise when we remove the associativity property. Finally, you can have a look at this: http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction.2012-09-07

1 Answers 1

5

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.