Not too long ago, whenever I was confronted with the expression, -x, and I was in a position where I needed to communicate it to someone verbally, I would say, "negative x", as opposed to "minus x", probably because "negative x" sounded more professional to me. This went on until a professor pointed out to me some of the problems with this usage of the word negative.

After all, whatever the object "negative x" is, it should at the very least be negative, right? However, objects written down as $-x$ (or $-g$ or whatever) are usually elements of a vector space or an abelian group, where there is usually no concept of negative element. Even when $x$ is a real number where the concept of negative element exists, the terminology is still bad (in fact it is probably worse, since in the case where $x$ is a negative number, "negative x" is not negative, but positive!).

I suspect that people regard the expression "negative x" as somehow emphasizing the fact that $-x$ is the additive inverse of $x$. This has other harmful effects. Just last year, acting under these pernicious influences, I caught myself telling linear algebra students about the "existence and uniqueness of negatives in a vector space"!

On the other hand the alternative, "minus x", is a straightfoward and unambiguous verbal description of the written form of the expression.

One might argue that this is a trivial issue which is hardly worth the effort of discussion, but mathematicians prize clarity, and if one alternative is clearly better than the other, why not stick to it? Another reason for using better terminology is that even though the underlying issues are trivial to us, they may not be clear to others. I distinctly recall being frustrated to tears at one point as a child, trying to understand how to add integers, and the phrase "negative one", etc. very well may have been a serious source of confusion. At the very least this discussion may be helpful for teaching mathematics to five year olds.

The expression "negative x" is not some fictitious straw man of my own construction. To the contrary, in my experience, it is the dominant verbal alternative. This turn of phrase even turns up in mathematical writing when authors decide not to assign a symbol to the mathematical object they are discussing. For example: "The 1-form is the negative of the differential of the function". It seems to me that the case for "minus x" is very strong, but I have nevertheless had a lot of difficulty winning people over to the "minus" side of the debate. Some argue that saying "negative x" is logical because it describes the process of obtaining -x from x by mutliplying x by -1, which is a negative number. However this strikes me as a convoluted way of constructing terminology, and the argument does nothing to address the potential for confusion.

I'll admit, there may be some arguments for "negative x" or against "minus x" which I haven't considered.

So what do you think?

"negative x" or "minus x"?