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I have been reading about multi-dimensional numbers, and found out that it's been proven that Octonions are the division algebra of the largest dimension. I was wondering why despite having infinitely many different dimensions of numbers, the only division algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8?

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See these three questions. –  J. M. May 16 '11 at 23:33
    
@J. M.: you are the link-master! Also, in light of "those three questions", would this be a duplicate topic? –  The Chaz 2.0 May 17 '11 at 0:06
    
@Chaz: I'm on the fence, and will thus let other people vote on it. –  J. M. May 17 '11 at 0:11
    
@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. –  lhf May 17 '11 at 0:37
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@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... –  J. M. May 17 '11 at 0:42

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up vote 7 down vote accepted

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 $2^{n-2} = n^2$.

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