Why use the notation (s, a, b, c) for quaternions instead of (s, v)?

[Septimra]

These are the notations of quaternions that i have seen:

q = s + v
q = (s, v)
q = s + ai + bj + ck

where s, a, b, & c are members of the reals

but why not use the notation of:

q = (s, a, b, c)

isn't it the same as the 2nd notation except it is clearer? So why does it take a quaternion to be defined as a 2-tuple over C² before that notation is possible?

 

[chiro]

Hey Septimra.

There are quite a few ways of defining a quaternion and one of them is actually the vector/scalar notation since the multiplication uses the cross product and scalar products to do this.

Aside from that you have your representation you posted above and you can use four variables with multiplication tables.

Basically as long as everything is mathematically consistent then it's all the same anyway.

I'd actually recommend looking at the vector/scalar representation if you want to understand what multiplication of quaternions does geometrically since you can visualize the cross product quite easily (in three dimensions).