Quaternions and co.

[moonlord]

I see that there are some equations in real that do not have solutions. So an I=sqrt(-1) was defined and all was fine. What are quaternions? I've heard about them in the 'Division by Zero' thread. Are there some equations in complex that do not have solutions? How are these numbers represented? What is the defined 'value' for their units (as I is defined as sqrt(-1))? I've also heard about octonions...

[thigle]

there's quite an amount of info about quats on the net.
they are represented generally as an ordered quadruple of numbers.

as for complex #s i² = -1 , for quats, ijk = i² = j² = k² = -1
sometimes, imaginary triplet is taken as vector and the real part as scalar.
or you can take this quadruple as ordered pair of complex numbers.

just google for quaternions, octonions... also mathWorld & wiki have entries.

or you can confortably start through doing physics with quaternions: forgotten path to new physics

or this, rather (un)clear one: quaternion math

have a good time with it, it's more complex than complex. hypercomplex some call it so if you google for 'hypercomplex', a lot of quaternion related websites pop up as well.

this one is basic for quats & EM hypercomplex.com

[moonlord]

Thanks very much! I've read the Wikipedia article on quaternions but it didn't say, for example, why were they invented. I hope your links make this and others clearer.

[houserichichi]

As far as I can recall, Hamilton was just trying his own intellect to see if it could be done (he was looking for a 3D extension to the complex numbers but couldn't find one...he had to bump to 4D). I could be wrong. Depending on your math background, Baez has a wonderful paper on octonions that may be readable. The Cayley Dickson process can be applied infinitely many times to produce other structures after the octonions (the sedenions are the next ones) but they're no longer division algebras (as was mentioned in another post).

The paper is here:

http://math.ucr.edu/home/baez/octonions/

[thigle]

on why they were invented, well, as houserichichi said, Hamilton was trying to find a minimal system of multiplication rules for rotations in 3-space, after the fact that multiplication by complex numbers in complex plane correspond to rotating.

another good one from Baez (maybe to read before the above link from house, which goes a lot deeper), an interview with Baez, math physicists, explaining curious quaternions in easily graspable way.

[bo198214]

As far as I can recall, Hamilton was just trying his own intellect to see if it could be done

Yes, I dont know either about any insufficiency of the complex numbers, that led to the construction of the quaternions.

[wendy]

Hamilton was trying to multiply three-dimensional numbers. He struggled with this for some good while, until he hit on the notion of using four.

W

[pat]

As others have mentioned, Hamilton came up with the quaternions in a successful attempt to describe 3-D rotations. Complex multiplication can model 2-D rotation. Quaternion multiplication can be used to model 3-D rotation.

I didn't see any answers to the original question though:

Are there some equations in complex that do not have solutions?

Well, I'm not sure what you'd call an equation exactly. Certainly the equation: 1/x = 0 has no solutions in the integers, rationals, reals, complexes, or quaternions. But...

There are polynomials with integer coefficients that have non-integer solutions. There are polynomials with rational coefficients that have non-rational solutions. There are polynomials with real coefficients that have non-real solutions.

However, every polynomial with complex coefficients has a complex solution.

Polynomials aren't quite as useful for quaternions since quaternions aren't commutative. You may have that (ax)² isn't equal to a²x². This makes polynomials messy. Rather than being able to write something like: a + bx + ex², you may have to write:
a + bx + xc + ex² + x²f + gxhx + xsxt + uxvxw.
That can be grouped a little better than that. But, you get the point.

People often focus on subsets then of left or right polynomials.... so a + bx + ex² is a left-polynomial (or is it right?) and a + xb + x²e is a right-polynomial (or is it left?). Every left polynomial with quaternion coefficients has a quaternion solution. Every right polynomial with quaternion coefficients has a quaternion solution.

There are some polynomials with quaternion coefficients that are neither right nor left (they are mixed...) which have no solutions in the quaternions.

The rationals are a ring-extension of the integers. The reals are a field-extension of the rationals. The complex numbers are a field-extension of the reals. The quaternions are a field-extension of the complexes.

There is no field-extension of the quaternions that provides solutions to the mixed polynomials. At least that's what this claims.
http://www.math.niu.edu/~rusin/known-math/95/quaternion.eq.
But, it's not clear to me what trouble comes from adding in those solutions. I'm thinking that what happens is that the whole thing collapses and you can show that everything equals zero if you append those solutions.