I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8?
Why is 8 so special?
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$\begingroup$
number-systems
octonions
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7See [these](http://math.stackexchange.com/questions/529) [three](http://math.stackexchange.com/questions/32100) [questions](http://math.stackexchange.com/questions/38604). – 2011-05-16
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0@J. M.: you are the link-master! Also, in light of "those three questions", would this be a duplicate topic? – 2011-05-17
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0@Chaz: I'm on the fence, and will thus let other people vote on it. – 2011-05-17
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0@J.M., I have to agree with The Chaz. I usually have a hard time searching for question here and I end up using Google with `site:math.stackexchange.com`. Any tips on searching? The default OR in searches here is not convenient and I don't think it searches comments. – 2011-05-17
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1@lhf: I did use Google (the built-in search is remarkably unhelpful); here the magic search-words are "quaternion" and "Frobenius". (Yes, Google can parse comments. Whodathunkit, eh?) Also, I happen to remember those three well... – 2011-05-17
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0http://www.youtube.com/watch?v=Tw8w4YPp4zM kinda both entertaining and informative – 2014-05-14
1 Answers
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You might be interested in Hurwitz's proof of his theorem (which is not as strong as Wikipedia's statement). Here is the original German and an English translation. The maximal $n$ turns out to be the solution of $2^{n-2} = n^2$.