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Comments on How $ijk=\sqrt{1}$?

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How $ijk=\sqrt{1}$?

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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).

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2 comment threads

As to why they're not commutative, think of them in terms of their application to describe rotation. ... (1 comment)
Quaternions are not complex numbers (1 comment)
Quaternions are not complex numbers
Peter Taylor‭ wrote almost 3 years ago

Quaternions are not complex numbers. The complex numbers can be seen as a subalgebra, but I'm not sure to what extent that is useful or helpful.