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Having taken some high school algebra, I got interested in complex numbers. Later on, I learned about quaternions. From what I've read:

$i^2=j^2=k^2=ijk=-1$

I took $i^2=ijk$ and divided $i$ from each side, which gives me this:

$i=jk$

If I do this with $j^2=ijk$ and $k^2=ijk$, I also get $j=ik$ and $k=ij$. So, the equation above can be rewritten:

$i=(ik)k$

Which simplifies to $i=ik^2$, which in turn simplifies to $i=i\cdot-1$ then to $i=-i$. Since $(-i)^2$ would be $1$, and $1=-1$ is obviously false, what did I do wrong?

I'm very new to this, just learning it for fun, so I'm sure I'm misunderstanding something obvious here. Thanks for any help!

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    Unfortunately commutativity fails in the quaternions, i.e. $ik\ne ki$. Explicitly we have $ik=-j$ and $ki = j$. – player3236 Sep 24 '20 at 20:58

1 Answers1

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Your error lies in

If I do this with $j^2 = ijk$ ...

The quaternion algebra is non-commutative, so any division should be either left-division or right-division.

For $i^2 = ijk$, you multiply both sides on the left by $i^{-1}$ and get $i = jk$.

For $k^2 = ijk$, you multiply both sides on the right by $k^{-1}$ and get $k = ij$.

But what do you do with $j^2 = ijk$? The $j$ on the right hand side is in the middle, so you do not directly get $j = ik$.

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