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: 
I think it's appropriate to lead the student(s) through answering their own objections with some leading questions - perhaps questions like 

"*When* is $-x$ *negative*?"

In particular it helps reinforce to students that *this is the sort of question they should be asking themselves*.

> 
> Why do students think this?
> 
> 
> 

I think it's because they're applying a "rule" that says something vaguely like "a minus in front of a number makes it negative" ... but it's an internalized idea they haven't articulated so they don't really notice the flaws in it.

Running into these sorts of errors is in some sense part of learning to think mathematically; it's okay to *make* these errors, as long as they're treated as something to recognize, and try to address. 

It's no good drilling someone for weeks on how not to fall off a bike -- they're still going to fall off when they're learning the skill. You can try to anticipate some things, but I don't think you should try too hard to make sure they never make an error. Errors are useful!

I've been doing mathematics for a long time; I make new mistakes almost every time I learn something new. Being able to identify and correct misapprehensions is part of getting there. I make fewer now than I did 30 or 40 years ago, but that's because I made enough errors to finally internalize things like "definitions matter".
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.