Comments on How $ijk=\sqrt{1}$?
Parent
How $ijk=\sqrt{1}$?
In the Wikipedia page, I can clearly see that $$i^2=j^2=k^2=ijk=-1$$
But if we consider them separately
$$i=\sqrt{-1}$$ $$j=\sqrt{-1}$$ $$k=\sqrt{-1}$$
So $$ijk=\sqrt{-1}\sqrt{-1}\sqrt{-1}=(-1)^{\dfrac{3}{2}}=\sqrt{-1}$$ It totally doesn't satisfy what I was looking for. In section, "Multiplication of basis elements" they assume that $ij=k$ and $ji=-k$. Even I can't find out why they are non-commutative (I know that matrices is non-commutative but can't find relation between matrices and complex number).
Post
The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=-1$.
How is that possible?!
Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=-1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
Ok, that's that. Then to directly answer the question
$$ ijk = (ij)k = kk = -1 $$
2 comment threads