[anaphylactic]

> There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided.

1. I think they meant "the only kinds of numbers constructed in this way". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. With octonions you've already lost associativity and commutativity, though.

[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

[OscarCunningham]

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.

[anaphylactic]

Yeah, octonion multiplication is alternative.

[gowld]

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...

[bonzini]

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

[OscarCunningham]

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.

[qubex]

That happens in general for matrices too.

[anaphylactic]

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.

[gjm11]

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.

[Iwan-Zotow]

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

[SubiculumCode]

An n-dimensional matrix of octonions

[nabla9]

Being composition algebras makes R,C,Q and O more interesting than others:

N(xy) = N(x)N(y) - N is called norm.

Without this property, you have zero divisors.

edit: throwawaymath uses better notation: |xy| = |x| • |y|

[jstanley]

What type of thing is N? What you wrote doesn't seem to make sense if N is just a constant, but I don't see how it makes sense if N is a function? Or maybe you didn't mean multiplication?

E.g. N = 3, x = 2, y = 3

N(xy) = 3(2 . 3) = 18

N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18

[throwawaymath]

That's not a subtraction sign, it's a hyphen. N is the norm of x, denoted by |x|. Technically the norm is a scalar-valued function applied to a vector, hence the functional notation N(xy) = N(x) • N(y).

So to be explicit, they're saying |xy| = |x| • |y| implies you cannot have a 0 divisor.

[Steuard]

N here is a function, not a constant. I was confused at first, but I think the "-N" in that post wasn't intended to mean "minus N" but rather ", where N is the function called...".

The "norm" of a number is more or less its absolute value: its size, it's magnitude. So "1" has norm 1, but "-1" also has norm 1, as does "i", and "-5" or "5i" or "4-3i" all have norm 5. So these four mathematical structure (R,C,Q, and O) all have the property that the norm of a product is equal to the product of the norms. Things get really obnoxious (or at least really unfamiliar) if you don't have that property.

Edit: Sorry for the repetition! I've got to remember to reload these pages before replying.

[donbright]

is there a proof that real numbers can be added?

[kgilpin]

As far as I know, addition is something you define (an axiom) in order to construct the real (or other) numbers.

https://sites.math.washington.edu/~hart/m524/realprop.pdf