Why do quaternions exist?
I want to believe they exist, but all I can think of are reasons they should not exist.
These are my reasons.
- The quaternions are defined by the following equation:
$$i^2 = j^2 = k^2 = ijk = -1$$
There are four equalities and three variables. There should be no solution.
-
Somehow there's a workaround for dilemma $1$. They exist.
So how do you define the quantity $ijkijk$?
Is it interpreted as $(ijk)(ijk)$, or $-1 \times -1$, or $1$?
Or is it interpreted as $(i^2)(j^2)(k^2)$, or $-1 \times -1 \times -1$, or $-1$?
$ijkijk$ can't be two values at once. -
I read that quaternions are not multiplication commutative. Why multiplication commutative? Why not multiplication associative or some other property?
-
Okay, so there is a reason for number 3.
Why can't I define a new set of numbers where the multiplicative identity doesn't hold?
Like: $1x = x + 1?$
I've been pondering these questions for a couple of days now, so I would really appreciate an answer.
Best Answer
They are not exactly variables, they are symbols. And we are just defining what happens when we apply a binary operator to them. You can always have more equations than variables, that doesn't mean you don't have a solution (even in your highschool algebra), it just means its possible there might not be a solution.
You are assuming the quaternions are commutative to make your argument here. They are not. You cannot rearrange terms. ij is different than ji.
Commutative means we can rearrange the terms. For example x * y = y * x. The quaternions are in fact associative which means (x * y) * z = x * (y * z). This is why it is not ambiguous to write x * y * z since the grouping doesn't matter. The quaternions are indeed associative. This is why $ijkijk$ is not ambiguous.
You can define whatever kind of structure you want. The quaternions are a Group which have the identity property. However there are other algebraic structures, such as Magmas, which impose less restrictions
If you are really interested in detailed answers, check out some abstract algebra books from your local university library.