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So after the quaternions came to my knowledge, I wonder if you could go any further with the complexity. Turns out you can with the octonions(8D numbers) and sedenions(16D numbers). But are there 32D numbers, 64D numbers, etc.?

Xavier Stanton
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    Yes. See https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction to see what properties are lost with each doubling. – GEdgar Dec 05 '18 at 01:32
  • I'm already not aware of any applications of the sedenions whatsoever (the octonions have some interesting applications to constructing exceptional Lie groups), so I don't particularly see a need to go any farther in this direction. In a related direction you could learn about rings in general, which have many applications: https://en.wikipedia.org/wiki/Ring_theory – Qiaochu Yuan Dec 05 '18 at 01:36

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