Why do division algebras always have a number of dimensions which is a power of $2$? Why do number systems always have a number of dimensions which is a power of $2$?


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*Real numbers: $2^0 = 1$ dimension.

*Complex numbers: $2^1 = 2$ dimensions.

*Quaternions: $2^2 = 4$ dimensions.

*Octonions: $2^3 = 8$ dimensions.

*Sedenions: $2^4 = 16$ dimensions.


The particular family of algebras you are talking about has dimension over $\Bbb R$ a power of $2$ by construction: the Cayley-Dickson construction to be precise.
