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It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.

I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified?

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These are often confused because students believe that $\sqrt{x^2}=x$, but actually $\sqrt{x^2}=|x|$. So: $\sqrt{x^2}=\sqrt{9}$ implies $|x|=3$, and so there are two possibilites: $x=3$ or $x=-3$. – rschwieb May 21 '13 at 21:06
up vote 8 down vote accepted

If you want your square-root function $\sqrt x$ to be a function, then it needs to have the properties of a function, in particular that for each element of the domain the function gives a single value from the codomain. If you take a function to be a set of ordered pairs, then each of the initial values of the pairs must appear exactly once.

So to be a function, square-root needs to be single valued; the multi-valued version is really a relation, at which point you might get into issues of principal values.

For convenience, the square root of non-negative real numbers is usually taken to be the non-negative real value, but there is nothing other than practicality to stop you from taking some other pattern. Such arbitrary choices can raise significant issues when considering, for example, cube-root functions defined on the real and complex numbers.

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For positive real $x$, $\sqrt x$ denotes the positive square root of $x$, by definition. Wikipedia agrees with me on this.

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If you solve an $\color{#F80}{\mathrm{equation}}$ containing an unknown variable say $x$; in your case such as: $$x^2=9$$ Then the equation has solutions given by $x=3$ and $x=-3$.

But if you are just given the $\color{purple}{\mathrm{expression}}$: $$\large\sqrt{9}$$ then the expression can only reduce to $3$ (Not $-3$).

So the number of solutions really simplifies to whether the radical in question belongs to an $\color{#F80}{\mathrm{equation}}$ or an $\color{purple}{\mathrm{expression}}$; where the latter will only take the principle (positive) root.

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