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~~In the [Wikipedia page](https://en.wikipedia.org/wiki/Quaternion#History), 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).

In the [Wikipedia page](https://en.wikipedia.org/wiki/Quaternion#History), 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 $$ijk=\sqrt{-1}\sqrt{-1}\sqrt{-1}=(-1)^{\dfrac{3}{2}}$$ 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).

quaternions