Why aren't square roots written between vertical bars (absolute value)?

Question

The root of a number $x$ is $y$ and $-y$.

So why aren't we writing either $|\sqrt{x}|$ or just $\sqrt{x}$ to determine if someone wants $-x$ or $x$?

Answer 1

There is only one square root: $\sqrt{}\colon \mathbb{R}_+\to\mathbb{R_+}$ is a function, defined as "$\sqrt{x}$ is the only non-negative number $t$ such that $t^2=x$."

You have that $-\sqrt{x}$ also satisfies $(-\sqrt{x})^2=x$, but it isn't the square root -- it's just another number that satisfies the same equation $t^2=x$ (but not the non-negativity condition).

Answer 2

It is a convention, and one which works. To make it as natural seeming as possible, I'd explain it as follows:

As we know:

OPTIONAL PREAMBLE: For each $a$ in the multiplicative group $ \mathbb{R}^+ $ there is exactly one solution (in that group) to $x^2=a.$ We call it $\sqrt{a}.$

In $\mathbb{R}$ there are, for each $a \gt 0$, exactly two solutions to $x^2=a$, one positive and one negative. We might choose to call the first $+\sqrt{a}$ and the other $-\sqrt{a}$ and this is correct. When we wish to refer to both we write $\pm \sqrt{a}.$ It is a universally accepted convention that $\sqrt{a}$ means $+\sqrt{a}.$ This does not conflict with the more symmetric practice of always recording the sign and it turns out to be quite convenient. Further defenses can be made for why it is "the right thing to do." However it is a convention and, perhaps, does not reflect an essential mathematical truth.

The same issue arises in other cases of finding inverses for functions which are not one to one. We restrict , if possible, to a convenient domain on which the function is one to one and then try to describe the full set of solutions in terms of the special one.

For example $\cos{\theta}=t$ has infinitely many solutions but only one in the radian range $[0,\pi].$ We call this unique (smallest non-negative) solution $\arccos{t}.$ The general solution is then easily described as $\pm\arccos{t}+2k\pi.$

In mentioning the group $\mathbb{R}^+$ in the optional preamble I was trying to ease through the use of the the bald $\sqrt{a}$ because "math." But it all makes sense without the preamble.