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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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up vote 3 down vote accepted

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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