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There are complex $\mathbb C$, quaternions $\mathbb H$ and octonions $\mathbb O$. Is there any higher dimensional generalization of them, such in the $\mathbb R^{16}$? Or why do we just study three kinds of numbers in Mathematics?

Any advice is helpful. Thank you.

gaoxinge
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    The sets of real numbers ($\mathbb{R}$), complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$) and the octonions ($\mathbb{O}$) are the only normed division algebras that can exist. For more information the wikipedia page is informative! – Thomas Russell May 27 '15 at 12:44
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    See here.for a description of what happens if you extend the reals. If you are not worried about topological closure, then you have very many things to play with. For example if $K$ is a number field (= a finite extension of $\Bbb{Q}$), then for any integer $n>1$ you have infinitely many non-isomorphic division algebras (= all the non-zero elements are invertible) A) with center $K$, B) of dimension $n^2$. "Topological completeness"makes the algebra a bit degenerate leaving quaternions as the only non-commutative associative division algebra – Jyrki Lahtonen May 27 '15 at 12:45

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Complex numbers are nice because they are similar to the reals in that that multiplication is associative and commutative. The quaternions have different algebraic properties as their multiplication isn't commutative. Multiplication in octernions is not even associative (so they don't form a field for example). There are some higher level generalisations of them but they are even less useful as they have even fewer nice algebraic properties that make study interesting and fruitful.

Dan Robertson
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Kervaire and Bott & Milnor independently proved in 1958 that the only four division algebras built on the reals are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$.

In the step between complex numbers and quaternions we lose commutativity. Between quaternions and octonions we lose associativity.

See also this review paper: https://arxiv.org/pdf/math/0105155.pdf

Ben Hocking
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marwalix
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