Mathematics question

[epepke]

I forgot some names in math, and I couldn't find what I needed from a google search, so I figured here would be a good place to ask.

1. Real
2. Complex
4. Quaternion
8. Octonion
16. ??
32. ??

What are the ??s? It's the progression of numbers started with Hamilton's discovery of quaternions.

And is there a name for the entire series? I've seen names for up to octonions.

[Dragonrock]

I like cheese :D

[Iconoclast]

Originally posted by epepke
I forgot some names in math, and I couldn't find what I needed from a google search, so I figured here would be a good place to ask.

1. Real
2. Complex
4. Quaternion
8. Octonion
16. ??

Sedenion comes next, don't know if there's a name defined for 32 dimensions, and 16 dimensions is more than enough for anybody.

Originally posted by epepke
And is there a name for the entire series? I've seen names for up to octonions.
All the numbers with more dimensions than complex numbers are collectively known as Hypercomplex numbers.

[Vorticity]

Originally posted by Iconoclast
All the numbers with more dimensions than complex numbers are collectively known as Hypercomplex numbers.
But it's not just a matter of dimensions, right? Otherwise you'd just be talking about ordinary vectors. There are also certain multiplicative rules between the components that have no analog for ordinary n-vectors.
Examples:

For complex numbers, you have 1 and i, with the rule that ii=-1

For quaternions, you have 1, i, j, and k, with the rules
ii=jj=kk=ijk=-1
ij=-ji=k
jk=-kj=i
ki=-ik=j

I've no inkling what the rules might be in higher dimensions, but I imagine it could be figured out from group theory, Pauli matrices, non-abelian, blah blah blah, etc... etc. :D

[Yahweh]

Originally posted by Dragonrock
I like cheese :D
With grits?

[Tez]

The octonians are the largest division ring (http://mathworld.wolfram.com/DivisionAlgebra.html) and this is why we normally stop the sequence there...

[epepke]

Originally posted by Vorticity

But it's not just a matter of dimensions, right? Otherwise you'd just be talking about ordinary vectors.

Right. "Hypercomplex" really isn't accurate enough, because sometimes it's used to describe just any bunch of numbers, sometimes only up to octonions, and sometimes to describe systems according to the differences from real and complex algebras rather than similarities (which include matrix algebras, etc.)

Each kind of number is defined from the lower order numbers. Quaternions are such that i, j, and k each individually work like the i in complex numbers, but together they work according to the rules you posted. Octonions are defined such that each of the triplets (i0, i1, i3), (i1, i2, i4) work like quaternions.

At each stage you seem to have to give up some more. At quaternions, you lose commutativity. At octonions, you lose associativity. At sedenions, you lose the division ring, but you still keep a multiplicative identity and inverse. You also lose the alternative property.

If I'm remembering properly, the 32-scalar number is the biggest you can have without losing the multiplicative identity and inverse. I don't remember what you lose at 32, probably power associativity.