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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.

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Have you seen the sedenions? As to why Hamilton came up with the quaternions, see this article, for instance. – J. M. Sep 7 '12 at 0:16
For an application of quaternions, you may be interested in en.wikipedia.org/wiki/Quaternions_and_spatial_rotation – Trevor Wilson Sep 7 '12 at 0:20
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You could have a look at this: math.stackexchange.com/questions/529/…. Also, sedenions arise when we remove the associativity property. Finally, you can have a look at this: en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction. – M Turgeon Sep 7 '12 at 0:35

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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.

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