What is 'above' the complex numbers??

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[moonsweater]

"Even more non-associative" is terrifying to read

[Bolpat]

From R to C, you lose the ordering, but you gain algebraic closure.

[Tehom]

For those who are curious, an easy way to see that polynomials over the quaternions may have infinitely many roots is to consider the roots of x^2+1. The complex numbers have roots {i, -i} because i^2 = -1 = (-i)^2. But notice that the quaternions bases {i, j, k} are all defined to square to -1, so all of them work. Combinations such as (i + j)/sqrt(2) square to -1 too, and in general any unit imaginary quaternion squares to -1, so x^2+1 has all of them as roots.

[ow]

Please make a video briefly going through the octonions and sionions showing how they're non associative!

[Morgyborgyblob]

I remember when I first found out about complex numbers, and how they could be visualised as extending the number line into a plane, was to ask if there was a third dimension. I'd heard about quaternions, but not the higher order spaces. I think I know what my evening's googling will be on!

[sb-qxew]

Though this process of dimension doubling (the so-called Cayley-Dickson construction) can be continued indefinitely, only the first four are what are called "normed division algebras"; roughly, field-like things admitting a sensible notion of length and multiplicative inverses. From the sedenions onwards, they're kind of useless (at least to my knowledge).

A theorem of Hurwitz establishes that these four are the only possible normed division algebras over the reals; this has some far reaching consequences. For instance, this is why the only cross products exist in 0, 1, 3, and 7 dimensions; even further afield, they give rise to Bott periodicity in stable homotopy theory.

I highly recommend that the interested see John Baez' excellent "The Octonions" for more detail!

[garysimpson]

Supposedly, Paul Dirac once stated that the quaternions were the simplest system that was sufficiently complex to describe QM.

[givrally]

I would say it's the other way around, the more properties you're willing to lose, the higher you can go.

[_mark_]

What does “even more non-associative” mean

[prosfilaes]

I had a problem with a teacher once over "ordered"; if you've studied more set theory and abstract algebra than fields, the concept that the complex numbers aren't ordered is confusing; in fact, that any set is not well-orderable is a big statement. (Said teacher declined to explain what he meant by ordered and instead mock me in class the next day.) "Ordered" meaning "orderable as an extension of the reals allowing negative and positive numbers to work under addition and multiplication as you'd expect" is not the first meaning that comes to mind.

[loganhodgsn]

I'd love to learn more about these groups!

[atellsoundtheory]

Since they don't generalize as a proper field, you might just as well consider the complex numbers as an algebra over the reals and then they generalize elegantly to Clifford algebras. I think this is a much better way to think of them, especially since the quaternions are also contained in the Clifford structure.

[Necrozene]

You sir, are a super excellent maths teacher. One day, I might lean on you to explain something. So far, you are doing great!

[shruggzdastr-facedclown]

I think that it's better to say "beyond" than "above"

[edindrevataj]

Does a^x=0, a>0 have solution in any of those numbers?

[darkestkhan]

Quaternions are nice and highly practical. They nicely describe 3D rotations, among other things.

[bluematter]

see because i have used these things (complex numbers, quaternions) only due to extra features, both having to do with working nicely with rotations (one in 2D and another in 3D)
i have always seen them as simply having more useful properties.
I have never considered how im loosing other properties i used to have along the way;
but this is such a neat thought

[henrikholst]

Is it really true we cannot order the complex numbers? Suppose you take z < w if |z| < |w| or if equal if arg z < arg w. Wouldn't that be a order? I think it's more to it something with continuity the order won't be useful.

[monkey]

And sedenions have "zero dividers"
And that definitally sounddms stanrange

[RGAstrofotografia]

And above we have the 32 Pathions, 64 Chingons, 128 Routons and the 256 Voudons!