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Comment by Maybestring | Hacker News Reader

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Maybestring·

>it's just that multiplication and division lose most of their useful properties.

Specifically they lose the property of not having zero divisors.

There exists sedonions a,b != 0 such that ab = 0

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Another (related) property that fails is that inverses stop being useful for cancellation. Inverses still exist, for every p there's a q with pq = qp = 1, but if you've got an equation ap = b you can't cancel to get a = bq, because we don't have associativity. The left hand side (ap)q doesn't equal a(pq), so you can't reduce it to a.

Of course associativity doesn't hold in the octonions either, but it holds just enough for cancellation to work.

Yeah, octonion multiplication is alternative.

Are there 32-onions that lack power associativity?

https://en.wikipedia.org/wiki/Power_associativity

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

No, power associativity is never lost, after sedenions the properties remain the same.

They don't remain exactly the same. The answers to this (https://math.stackexchange.com/questions/641809/what-specifi...) question provide some interesting starting-points.

That happens in general for matrices too.

Yes, and this is a bit obvious, but reals, complex numbers, split complex numbers, quaternions, octonions, sedenions, can all be represented as matrices of the appropriate form.

That's at most sort-of-true. It's not possible to represent octonions by matrices of numbers in such a way that multiplication of matrices corresponds to multiplication of octonions, because matrix multiplication is associative and octonion multiplication isn't.

can't be true, unless there is some special matrix multiplication rule - afaik, standard matrix multiplication is associative

An n-dimensional matrix of octonions

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