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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.

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2 Answers

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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:

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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.

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