# The plus minus sign after take a root of terms [duplicate]

It's basic question, but sometimes might confuse me.

Why do we adding the plus minus sign $$(\pm)$$ after take a root of something? And why sometimes we just write the plus term only. I can't give a specific example that explain what i mean is, but i know if this question is clear.

Sorry for my stupid question, maybe i have some misconceptions about it.

• If $x^{2}=y$ then $x= \pm \sqrt y$ in general. If I already know that $x \geq 0$ then $x=\sqrt y$. Commented Jun 20, 2019 at 7:38
• $x^2=k$ has two solutions for $x$ when $k$ is positive. But you want the square root function to be a function so $\sqrt k$ only gives one value. It depends on which situation you are in Commented Jun 20, 2019 at 7:57
• I am not sure what kind of examples you have in mind. I can think of the following. The real numbers have a natural ordering, so when taking the square roots of $2$, say, one decides that $\sqrt{2}$ is the positive one, hence one writes $\pm \sqrt{2}$ to denote one or the other of the roots. The complex numbers, on the other hand, are not an ordered field, so $\sqrt{-5}$ may be taken to denote any of the two square roots of $-5$. Commented Jun 20, 2019 at 7:58

The symbol $$\sqrt x$$ usually denotes the square root function, which is defined as the positive root of the equation

$$y^2=x.$$

Hence, the solutions of this equation are

$$y=\sqrt x$$ and $$y=-\sqrt x,$$

which is compactly written as

$$y=\pm\sqrt x.$$

There are several uses:

• The statement "$$x=f(y)$$ or $$x=f(-y)$$" is abbreviated as $$x=f(\pm y)$$. For example, the quadratic formula to solve $$ax^2+bx+c=0$$ does this with $$f(y)=\frac{-b+y}{2a},\,y=\sqrt{b^2-4ac}$$.
• At other times, $$\pm$$ is used to make two statements simultaneously; for example, $$(u\pm v)^2=u^2\pm 2uv+v^2$$ means "$$(u+v)^2=u^2+2uv+v^2$$ and $$(u-v)^2=u^2-2uv+v^2$$".
• Occasionally you'll see $$\mp$$ meaning "the opposite of a $$\pm$$ sign elsewhere". For example, we can combine the facts $$u^3+v^3=(u+v)(u^2-uv+v^2),\,u^3-v^3=(u-v)(u^2+uv+v^2)$$ as $$u^3\pm v^3=(u\pm v)(u^2\mp uv+v^2)$$, meaning the two $$\pm$$s match each other but not the $$\mp$$.
• And sometimes, you'll even see $$\pm_1,\,\pm_2$$ referencing two independent signs, so a sign determined from them by multiplication would be $$\pm_1\pm_2$$, or the opposite would be $$\mp_1\pm_2$$. (You might see this in the roots of quartic equations one day.)

As for how a $$\pm$$ sign becomes a plain old $$+$$ or $$-$$, that's when you can deduce the sign somehow. For example, if you compute the golden ratio by solving $$\varphi^2-\varphi-1=0$$, the fact that $$\varphi>0$$ implies it's $$\frac{1\color{blue}{+}\sqrt{5}}{2}.$$I suspect most if not all of the examples of this you've seen set the sign to $$+$$, but sometimes it turns out to be $$-$$. For example, when you find the Catalan numbers' generating function, an argument about its behaviour for small $$x$$ reveals the correct expression is $$\frac{1\color{blue}{-}\sqrt{1-4x}}{2x}.$$(I hope you can see the colour of that $$\color{blue}{-}$$ sign.)