The definition I see everywhere for the absolute value of a complex number is:
Let $z=x+iy$ then $|z| := \sqrt{x^2+y^2}$.
But the square root operation is multivalued in complex analysis. So while this does show that the absolute value is a real number, it does not in fact indicate which square root we want. Of course it should be the nonnegative one, but what is the best way to express that? Would I just need to note "$|z| \ge 0$" in the definition? That's a little less than satisfying.
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$\begingroup$We usually take it as a convention that $\sqrt{x} \geq 0$, but if you wanted to make it explicit, you could say something like "$|z| : = \sqrt{x^2 + y^2}$, where in the definition we adopt the convention that $\sqrt{r} \geq 0$."
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