I am confused about this. $\sqrt{4}$ is $2$ or $\pm 2$? If $a \in \mathbb{C}$ then will there be the $\pm$ sign in $\sqrt{a}$? when this plus minus thing arrives?
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For real numbers, there is a convention that always the positive square root (If $x>0$) is meant. I think there is some convention in $\mathbb C$ as well – Peter Jan 14 '17 at 15:43
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$\sqrt 4$ is always $2$, by definition. But if you know that the square of something is $4$, then that something is either $2$ or $-2$, and without more information it's impossible to tell which. That is the reason you see $\pm \sqrt{\vphantom x \hphantom x}$ appear, for instance in the quadratic formula; in the derivation of that formula, we come across a point where we know that $(2ax + b)^2 = b^2 - 4ac$. We don't know whether $2ax+b$ is positive or negative, and we resolve that by saying $2ax + b = \pm \sqrt{b^2 - 4ac}$.
With complex numbers, you should use $\sqrt{\vphantom x \hphantom x}$ only sparingly, and if you decide to use it, then always with a $\pm$ in front. This is because there is no distinguished square root for complex numbers, and we must therefore always carry with us both until we can decide which one to keep. The exception is, of course, that the real or imaginary part may have square roots in them, for instance $\sqrt3 + i$.
Arthur
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