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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.

amWhy
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Asinomás
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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. ain't a mathematician Sep 07 '12 at 00:16
  • For an application of quaternions, you may be interested in http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation – Trevor Wilson Sep 07 '12 at 00:20
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    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. – M Turgeon Sep 07 '12 at 00:35

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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.

Michael Joyce
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