Is it OK to write $z^{1/2} = \pm z^{1/2}$, where $z \in C$?

Question

In particular, is it OK to write $z^{1/2} = - z^{1/2}$, where $z \in C$?

Answer 1

No, not at all. First of all, $z^{1/2}$ is ambiguous, since every non-zero complex number has two square roots. So, unless you explain first how you are choosing a square root, don't use $z^{1/2}$ and, for the very same reason, don't use $\sqrt z$.

Besides, in $\mathbb C$, $a=-a\iff a=0$.

Answer 2

Usually, when we write an algebraic expression such as $z^{1/2}$, we are assuming that it has a unqiue value. For example writing something like $z^{1/2}=i=-i$ will give you some weird looks, to say the least. Sure, what you mean by this could be that $i$ and $-i$ are the values which satisfy $z^2=-1$ (which is true), but this is not how you denote it. For the same reason $z^{1/2}=-z^{1/2}$ is not acceptable notation.

Furthermore, as José states, writing $z^{1/2}$ is already frowned upon as it is not clear which square root of $z$ is intended (since there are two possible values). So to even use this notation, be sure to define it first (e.g. as the root with the smallest argument).