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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.

Mathmo123
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user516076
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    If $x^{2}=y$ then $x= \pm \sqrt y$ in general. If I already know that $x \geq 0$ then $x=\sqrt y$. – Kavi Rama Murthy Jun 20 '19 at 07:38
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    $x^2=k$ 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 – Henry Jun 20 '19 at 07:57
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    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 $-5$. – Andreas Caranti Jun 20 '19 at 07:58

2 Answers2

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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

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

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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.)

J.G.
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