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You can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on. https://en.wikipedia.org/wiki/Cayley%E2%80%93Dicks
by leereeves 6y ago
You can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on.
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...
That's different from extending 1d to 2d, 3d, etc as vectors.
- gspr 6y agoI know. But these are not as fundamentally important as the complex numbers.
- AnimalMuppet 6y ago"And so on"? I thought octonions are where it stops? At least, anything past the octonions is not able to be a normed division algebra, if I understand correctly.
- leereeves 6y agoIt doesn't stop, but as you said, the next step (sedenions), and all steps after that have non-trivial zero divisors and aren't division algebras.
- gspr 6y agoExactly. This is what I was hinting at when critical at people who say complex numbers are "just 2d reals with fancy multiplication". It's true, of course, but it makes it seem as if the 2 in "2d" is unimportant/arbitrary.