Complex number (square) root notation.

Question

A mathematician told me that the notation $\sqrt{a+bi}$ isn't used, instead we use $w=z^2$ and substitute. Is this correct? If yes, is there any particular reason we don't want imaginary numbers under a root sign?

Answer 1

In general the equation $$z^2=w$$ has two (complex) solutions $z_1$ and $z_2=-z_1$.

In the case $w$ is real and positive, the solutions are real. Thus exactly one solution is positive and one solution is negative. It is then possible to assign $\sqrt{w}$ to the positive solution.

Return to the general complex case, $\Bbb C$ cannot be equipped with an order as $\Bbb R$. The two solutions $z_1$ and $z_2$ are now more equivalent. It is then not natural, even impossible, to assign $\sqrt{w}$ to either of the two solutions.

Answer 2

To write $\sqrt{z}$ for one of the two, or for both, complex square roots of complex number is perfectly fine, apart from the inescapable ambiguity. It certainly expresses the intent. In many contexts, further clarification would be needed to specify which of the two square roots, perhaps depending on how $z$ moved along a path or within a region.