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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?

Antonio Vargas
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wrongusername
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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
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    I would say this can answer your question. See also this. –  Mar 11 '11 at 09:49

5 Answers5

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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.

Henry
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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.

TonyK
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    I agree that there is a strong convention that $\sqrt{x}$ means the positive square root. But I think your wording overstates the position. Wikipedia is sometimes wrong, and anyway there is no authority in maths that lays down definitions. – almagest May 14 '21 at 13:01
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    @almagest, any other convention would be unworkable, now or when this answer was posted :-) – TonyK May 14 '21 at 14:42
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The radical sign '$√$' means we are taking the positive square root of given equation

if we simply say taking square roots on both sides, then we apply a '$±$' before radical('$√$') sign, as I said '$√$' sign means positive square root, so in order to get negative one also we apply that '$±$' sign.

as you can see '$(±√x)^2$' gives result as $x$, i.e $(+√x)(+√x)=x$ and $(-√x)(-√x)=x$

The simplest way to understand this is by the following expression

if $x^2=9$

taking square root on both sides

$±√x^2=±√9$

$$±|x|=±|3|$$

it follows

$$+|x|=3$$$$-|x|=-3$$

in order to define $√$ positive, mathematicians added | |, this is called modulus function, which makes everything positive So x=3 or x=-3 so $x=±3$ or we can say $x=±√9$ as I said again $√9$ is always positive

  notice I have used word **Square root** not the symbol, means we are taking both positive square root and negative square root

but when we say $√x^2$,notice here is no $±$ symbol,so here, it is asked for the positive square root only

Conclusion: We conclude that $√$ is defined to be positive

you can also see this in Quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$$

there is written $±$ in order to include negative root too!

hope it helped you......

Dheeraj Gujrathi
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    Hmmm. I don't understand why this is flagged as a low-quality answer. It is correct that: (1) a positive real number has two square roots (each the negative of the other); (2) there is a strong convention that the square root sign refers to the positive square root. That is a complete answer to the OP's question. The other answer (showing as +30 votes currently) is strictly correct, but surely unhelpful, in pointing out that one can define any symbol to mean anything in maths. The convention on the square root sign is very widely accepted, so deliberately flouting it just causes confusion. – almagest May 14 '21 at 12:46
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    Top tier answer. – Orange Cat Aug 30 '23 at 03:42
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Consider the similar equation $x^4 = 9$. This one has four solutions. The most obvious solution is $\sqrt 3$, which is roughly 1.732. Another solution is $-\sqrt 3 \approx - 1.732$. The same as the previous solution, just multiplied by $-1$.

There are two other solutions: $x = \sqrt{-3}$ and $x = -\sqrt{-3}$, which are roughly 1.732, the former multiplied by $i$ and the latter multiplied by $-i$. These are all the same distance away from 0.

If you just need one solution, you might as well take the solution that is a positive real number. If you need the other solutions, just multiply the solution you have by units other than 1.

So to solve $x^2 = 9$, the solution $x = 3$ might be enough, but if you need the other solution you just multiply the previous solution by the unit $-1$.

Robert Soupe
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The positive value is taken because it's more valuable in general, since the domains of elementary functions which aren't defined on the entire real line are often the positive or non-negative numbers. For instance, taking the positive value lets you say $\sqrt{\sqrt{x}} = \sqrt[4]{x}$ or $\ln{\sqrt{x}} = \frac{1}{2} \ln{x}$.

Zemyla
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