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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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    $\begingroup$ This square root means the positive real value. $\endgroup$ – Akatsuki Feb 21 '16 at 1:22
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    $\begingroup$ $x$ and $y$ are real numbers. $\endgroup$ – Future Feb 21 '16 at 1:23
  • $\begingroup$ I know that. So we should just use more than one definition of the square root symbol in complex analysis depending on whether the number under the square root is complex or real (which of course is just a complex number itself)? $\endgroup$ – user316338 Feb 21 '16 at 1:24
  • $\begingroup$ The interpretation of "modulus" is that it is the distance of $ \ z \ $ from the origin; consequently, we take the positive square-root of the sum of squares of real numbers, as the Pythagorean Theorem indicates. There isn't any hidden subtlety intended in the definition. $\endgroup$ – colormegone Feb 21 '16 at 1:30
  • $\begingroup$ No. As for the real numbers you could write $\sqrt 4 =+-2$ you know that the function $\sqrt x$ is the inverse of the square function with domain $\mathbb{R}+$ then $\sqrt 4=2$. In the complex numbers it's true that the equation $x^4=3$ has four different solutions, but still by $sqrt[4]{3}$ you just mean the real positive value. $\endgroup$ – Nicolò Feb 21 '16 at 1:33
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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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