Question: [This](https://matheducators.stackexchange.com/questions/5794/how-to-explain-that-a-negative-number-multiplied-by-a-negative-number-is-a-posit) recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative number, and negative numbers don't have square roots. This is in the context of a calculus class for first-year college students, so these are students who ostensibly know algebra (and most of whom have taken a year of calculus already). (In the context of this class, characterizing negative numbers as not having square roots is correct; they know about complex numbers, and also know they're off-topic.)

My experience so far has been that these students can handle it after being reminded that it's possible that $x=-2$, so I can address it in an ad hoc way when it comes up, but I suspect it indicates a bigger conceptual gap. So my question is:

1. Why do students think this?
2. What algebra should I be reviewing to head this off before the specific case comes up?

Answer: 
Instead of thinking about $-x$, consider the expression $(0-x)$.

Having (hopefully) being exposed to expressions of the form $(a-b)$ for a long time, where $a$ and $b$ can be arbitrary reals, there's no reason to assume that the value of the expression is positive (or negative) without further information.

Therefore, when they see $\sqrt{-x}$, they can now think of $\sqrt{-x}$ $\equiv$ $\sqrt{(0-x)}$, and realise that there is a real root where $x$ has a negative value.
Pronouncing $-x$ as "negative $x$" leads to confusion because "negative" as an adjective means "less than zero". A red car is red; an even number is even; but negative $x$ might not be negative.

If math had evolved differently, we could have been using different symbolism today. Here's how the two symbolisms would translate:

$\begin{array}{rcl}
2 & : & \overset{\rightarrow}{2}\\
-2 & : & \overset{\leftarrow}{2}\\
-x & : & \overset{\rightleftharpoons}{x}\\
x-y & : & x + \overset{\rightleftharpoons}{y}
\end{array}$

You could show this to your students, tell them they can mentally translate between the two if it helps them, and pronounce "$-x$" / "$\overset{\rightleftharpoons}{x}$" as "$x$ flipped" to emphasize that it's what you get by flipping the number line.