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: 
The following "definition" given in the Glossary of the Common Core State Standards for Mathematics (2010) give you an idea why students think this:

> 
> **Integer**. A number expressible in the form a or – a for some whole number a.
> 
> 
> 

The problem is not about algebra alone, rather it is about disconnection of teaching of algebra and negative numbers in schools. It has also a long history in which some great mathematicians of the past knowingly or unknowingly avoided negative values (particularly, as input) for an algebraic letter. The following is an interesting passage from Euler (taken from an under review paper of mine that is about the exact same issue you asked).

> 
> The question which we have hitherto considered lead all to an equation
> of the form $ax+ by=c$, in $a$,$b$, and $c$, represent integer and
> positive numbers, and in which the values of $x$ and $y$ must likewise
> be integer and positive. Now, if $b$ is negative, and the equation has
> the form $ax-by=c$, we have questions of quite a different kind,
> admitting an infinite number of answers.
> 
> 
> 

At the first glance, it seems Euler is being careless when he is saying "if $b$ is negative, and the equation has the form $ax-by=c$..."; if $b$ is negative in the sense that we understand it today, the equation $ax-by=c$ would be like the equation $ax+dy=c$, in which $a$,$d$, and $c$, represent integer and positive numbers, and Euler has already considered such an equation. But, Euler is not neglectful: when he is saying that "if $b$ is negative", he uses the letter $b$ as a label denoting the coefficient of $y$, and then the letter $b$ in the equation $ax-by=c$ is a parameter that represents integer and positive numbers.

If there is a way to bypass a negative number as an input of a letter, people find it and use it! Thus, a short answer to your second question would be to review the algebraic nature of negative numbers.

**Added.** The following Figure taken from a paper of Warwick Sawyer, [The Importance of the Unbelievable](http://www.marco-learningsystems.com/pages/sawyer/unbelieve.html), beautifully shows the importance (though often taken for granted) role of negative numbers in our most basic algebraic conceptions.I often use it in my classes to bring the unifying nature of accepting negative numbers to the fore.

![enter image description here](https://i.stack.imgur.com/S0RoT.gif)
I'd conjecture that classes have not given sufficient time to the specific translation skill between the English and Algebra languages, especially the meaning of relation symbols like equals (=). Two days ago I had to spend a few minutes clarifying this in my College Algebra class.

* Q: "Algebraically, how would one say that $c$ is a negative number?"
* A: "$-c$" (this from one of my best students)

This brings up a few problems. Primarily, this response isn't any kind of assertive statement; they fail to intuit relational symbols (=, <, >, etc.) as the necessary verbs in a sentence no matter how much I say, drill, or grade on it (obviously, "$-c$" is merely an expression, equivalent to a sentence fragment). Generally when I first meet students they place or omit equals signs pretty randomly on the page, as if it's mostly a decorative element, and are frequently bewildered at my grading for this on tests. 

When I write that the English "c is a positive number" is equivalent to $c > 0$, then most of my students can correctly answer that "c is a negative number" is equivalent to $c < 0$. For at least one weaker student I had to go over this again after class, but she did seem to have an "aha" moment after I wrote down an explicit table (English vs. Algebra columns).

In summary: Practically none of my students can intuit that "c is a negative number" would be written $c < 0$ without some direct instruction on the issue. Without that, their first guess is indeed that $-c$ has that meaning.