It seems that either numerical equivalence is not an equivalence relation or complex numbers do not have an inverse.
We know that $-1 = i \cdot i$
We also know that $-1 = -i \cdot -i$
If we assume "=" to be an equivalence relation we can get $i \cdot i = -i \cdot -i$
And $i \cdot i = -(i \cdot i)$
If we assume $i^{-1}$ exists, then so does $(i \cdot i)^{-1}$
Thus $ (i \cdot i) (i \cdot i)^{-1} = -(i \cdot i)(i \cdot i)^{-1}$
And $1 = -1$, a contradiction.
But this can't be right, can it?