I will be a teaching assistant in an introductionary course in Calculus in a university next semester. I am looking over the section that covers complex numbers, and allthough the question is a bit pedantic, I would like to offer some rigorous reasoning if the students are curious:
In an exercise the students solve a set of linear equations of the type $2iz = 3+4i \text{ for } z \in \mathbb{C}$, where $z$ is the variable of interest. After we devide by $2i$ and multiply both enumerator and denominator of the R.H.S. with $-2i$, we find a number that solves the equation; $z = 2 - \frac{3}{2}i$. How can we now be certain that we have found all solutions to the original equaion? Should one check that the multiplicative inverse is unique in $\mathbb{C}$? After all, after polar form is introduced, one realizes that a complex number can have several representations, and might begin to wonder if this can lead to otherwise strange behaviour.
Any idea on how to rationalize this in a simple, but exact way, without having to dive into abstract algebra?