I recently discovered quaternions and quickly stumbled across a to me contradictory statement. This contradiction already seems to lie in the definition (Hamilton's Rules): $$i^2=j^2=k^2=ijk=-1$$
My issue is with the statement $ijk=-1$ If I just multiply the three squares I get something as follows: $$i^2j^2k^2=-1$$ $$(ijk)^2=-1$$
Now if I try the same thing from the second equation $ijk=-1$ I get a somewhat contradictory statement: $$ijk=-1$$ $$(ijk)^2=1$$
All of the operations should be legal as far as I'm concerned but this makes little sense to me. Any explanation is greatly appreciated!
Alright. That was very simple. I was not following legal algebraic operations. $$(ijk)^2=-1$$ does not follow from $$i^2j^2k^2=-1$$ because $(ijk)^2 = ijkijk \ne i^2j^2k^2$