[paulddraper]

You can actually go as far as you want to with the Cayley–Dickson construction [1] of algebras.

1. Complex numbers have associativity and communitivity of multiplication. (That is, (ab)c=a(bc) and ab=ba).

2. Quaternions have associativity but not communitivity.

3. Octonions have neither.

4. Sedenions [2], trigintaduonions, and not associative, commutative, nor even alternative [3]. (Alternative is associative specifically when the middle value is equal to one of the other's; i.e. a(ab)=(aa)b.)

[1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...

[2] https://en.wikipedia.org/wiki/Sedenion

[3] https://en.wikipedia.org/wiki/Alternative_algebra

[pflats]

Thanks for writing this! I referenced it in a reply upthread for why I like the equation.

https://news.ycombinator.com/item?id=22204995