"Big" Conjugates

Which parts of the above are imaginary? Or do I not get what you're doing?

Ok sorry, when I started writing the post I had a different idea in mind than when I finished :p

Ok, so, to clarify, I'm finding the conjugate of a 3d complex number, i and j both being the square root of -1, but i is on the y axis and j is the z axis. It should be this:

a + bi + cj

So then you'd multiply it by this:

a - bi - cj

and would end up with this:

aa - bibi - cjcj - 2bicj= aa + bb + cc + 2bc

and that was confusing me for a while, I wanted to get rid of the 2bicj, but if i and j are both sqrt(-1) then they would cancel themselves out, so basically that's a solution (I think? Maybe someone can find something wrong with it?)

Well, if i == j, then

a + bi + cj == a + (b + c) i

so the conjugate is

a - ( b + c ) i

But, I don't think that's what you're really after.

I think the idea of a 3D complex number is flawed, because there aren't two ways of saying -1^1/2; so if two of your dimensions are in terms of that, they'll project onto each other. (i == j)

The binary notion of complex numbers is a consequence of things either being imaginary or real. There isn't a true third class of numbers that has a useful analog to real/imaginary, so a 3D complex number doesn't mean much in terms of conjugation--otherwise, it's just a 3-tuple.

I know, the concept sucks but maybe it will work with the way I'm doing it :D

This is the area of quaternions.

You have 1,i,j,k

i*i = j*j = k*k = i*j*k = -1

This is the definition for i,j,k

From this you can find that i*j=k, j*k=i, k*i*=j, j*i=-k, k*j=-i, i*k=-j

Nice :)

Multiplication is not commutative, but associative and distributive for addition.

So you get i*j = k

It works only in 3 ways:

1. real numbers, no imaginary units, this is the real number set

2. real part and one imaginary part, this is the complex number set

3. real part and three imaginary parts, this is the quaternion number set

Don't search other combinations, you won't find any without contradictions, this is proven.

Wow, thanks szinkopa, good post. I'll read more about these quaternion deals.