# Quaternions - Meaning and multiplication

1. Jun 6, 2010

### david.aloha

All right, I've been doing a lot of reading on quaternions, and while I think I understand how to use them, I'm still very confused as to why certain things are defined the way that they are. First question:
Why, when multiplying the imaginary portions of quaternions do you get these "rules":
i*i = j*j = k*k = -1
i*j = k, j*i = -k
j*k = i, k*j = -i
k*i = j, i*k = -j

I understand why i*i = -1, but why does an imaginary number times an imaginary number sometimes equal the negative of another imaginary number? I can't seem to find any underlying reason for this in anything I've read - it's just stated.

Second question:
Why use imaginary numbers at all if you're just defining a 3-vector plus a scalar (at least in terms of its use for 3D rotation which is the main context I've viewed it in)? What is complex or imaginary about it? The imaginary parts each define an axis in the same way as x, y, and z in a 3-vector - I'm just struggling to find what's different or special about adding the imaginary numbers (which, as I said above, also confuse me by only sometimes acting like imaginary numbers in multiplication).

I've spent a few days now looking through many different guides to quaternions, and none of them seem to answer these questions. Any help or clarification would be greatly appreciated. I'm the kind of guy who needs to know WHY something works, not just HOW it works, and this lack of why is driving me nuts.

2. Jun 6, 2010

### Hurkyl

Staff Emeritus
We can define multiplication pretty much however we want to get an algebra. That particular definition is nice, because multiplication is associative and every nonzero element has an inverse.

My limited knowledge of history is that quaternions inspired the invention of the dot and cross products.

3. Jun 6, 2010

### Martin Rattigan

Which possibly makes it sound more arbitrary than it might otherwise be regarded - see also http://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras).

4. Jun 7, 2010

### david.aloha

While large parts of the link you gave were a bit beyond me, I think I understand the significance behind why Quaternions are they way they are as a sort of base for "associative division algebras" (along with real numbers and complex numbers). I was reading this letter from Hamilton on how he arrived at Quaternions and I'm seeing a little better the process behind how he came to those rules:
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/QLetter/QLetter.pdf

I haven't read the entire letter yet (takes a long time when I'm really trying to formulate all of what he's saying) but I think I have a better understanding now than I did previously. Thanks for the responses.