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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"?

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I seem to recall that "minus $x$" is preferred by most mathematicians and math educators. To me the issue does not seem serious: surely "negative $x$" means $-1 \cdot x$ and not $-|x|$? –  Pete L. Clark May 15 '10 at 0:17
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I think "minus" should be used to denote an operation. I usually read "-x" as "the opposite of x", but that might be too much of a mouthful. –  Steven Gubkin May 15 '10 at 0:26
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This question is a good one for a blog post, but not for MO --- it is at heart a discussion question, and as such I vote to close. –  Theo Johnson-Freyd May 15 '10 at 3:17
    
Shouldn't this be community wiki at any rate? –  Regenbogen May 15 '10 at 6:52
    
The question has been closed. This is not a question with a well-defined correct answer, and the 12 answers given so far are quite similar to each other. This seems to be essentially an opinion poll, and MO is not well-suited to such things. –  Pete L. Clark May 15 '10 at 15:37
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closed as not constructive by Harry Gindi, S. Carnahan, Theo Johnson-Freyd, Robin Chapman, Pete L. Clark May 15 '10 at 15:31

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9 Answers

One way to defend "minus x" against "negative x" is to say that "minus x" is short for "0 minus x". There is a historical precedent for this. In Bombelli's L'algebra of 1572 (the book in which he successfully explains the occurrence of imaginary numbers in formulas that represent real roots of cubic equations) he writes $-1$ as 0 m 1.

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And I'm surprised that the multiplicative equivalent of this does not exist: nobody pronounces $1/a$ as "over $a$" (as opposed to "one over $a$" or "$a$ inverse"), or writes "$/a$" (unary use of the division operator) although that would be no more ambiguous than writing $-a$ instead of $0-a$. –  Marc van Leeuwen Sep 4 '13 at 11:42
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There was a terrible controversy in the 60's concerning American school mathematics. "New Math" adopted the term "negative 2" instead of "minus 2," apparently because it was believed to be confusing to use the same name for the unary operator as for the binary. For all I know, there may actually have been data to support the belief. Unfortunately, some educators, possibly the same ones, adopted the term "minus number" instead of "negative number." Thus the following hideous sentence would be correct: "Negative 2 is minus."

An professor of mathematics education insisted to me in 1965, that although "negative 2" was meaningful, "negative x" was not. I pretended to understand.

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funny; did they not to write, say ñ 2 for, you know, "negative 2", on the grounds that it's less confusing to have different symbols for the unary minus (ahem, "negate") operation and the binary subtraction operation? –  some guy on the street May 15 '10 at 3:57
    
(I'm on the edge of screaming at where markup, html, etc. gets used and where it doesn't, btw...) –  some guy on the street May 15 '10 at 3:58
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I always think of it as a British vs American thing. British students say 'minus 6', and all the American students I've taught say 'negative 6'. Presumably, as Carl Weisman says, this can be ascribed to 'New Math'.

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To me, "minus x" sounds awkward because operators are grammatically close to verbs. "Negative", on the other hand, is clearly an adjective, so even with the given defects, I think "negative x" sounds much better than "minus x".

That said, I echo Steven Gubkin's comment above; if I'm worried that "negative x" will be misinterpreted, I use "the opposite of x". (I don't know how other people were taught, but I learned the the numerator for the quadratic formula began with "the opposite of b...")

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I always say "minus $x$", but then again that's because in my native language (Spanish) that's how we say it.

In any case, $-x$ is the result of applying a unary operation to $x$. Hence it sounds strange to me to use the adjective "negative": surely "negative" is not describing an attribute of $x$ nor specifying it further, but actually changing it altogether!

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It's "minus $x$". I never heard the verbose alternative "negative $x$" until well into my adult life, and I suspect it's an Americanism.

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The issue with "minus x" is students often see

-3x=9

think "minus 3 x equals 6" and then decide to add 3 to both sides of the equation to get x.

Students really do interpret minus as subtraction. When reading -3x=9 they usually (if mentally thinking subtraction) reach for x-3=9, not (0-3)x=9.

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The thing with that is that bad habits in reading an unambiguous formula shows at least a defect of learning in the student; confusing use of vocalized "minus" and "negative" may be an actual obstacle to learning, more than terseness of formulae. For what it's worth, I've had calculus tutees who would look at (0-3)x=9 and still proceed to x=9+3. I mean, yeesh! –  some guy on the street May 15 '10 at 4:06
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"negative x" is "that which negates x [via addition]", which makes sense no matter what the nature of "x".

As for "positive" ... I don't know exactly what the "New Math" designers were thinking, but my sense is that the word is essentially a pre-emptive reaction to introduction of the term "negative". "Positive 3" is just the 3 you've known since you started counting, the 3 that describes how many elephants can be put here ("posit" = "put"), the "regular" 3, but with a fancy-sounding name applied in explicit contrast to "negative 3", the new-fangled number that cancels-out that familiar one.

When I'm near a mirror, there's (left-handed) "me" and (right-handed) "reflected-me". If I want to discuss the two of us, I might occasionally need to emphasize that the lefty is me --this one, here!-- so I might be inclined to use a phrase like "actual-me" or "original-me". In this sense, "reflected" is like "negative", and "actual" is like "positive" ... a word I apply merely help make a distinction ... an explicit redundancy, given my natural bias toward half of the "people" near the mirror.

Tying the analogy back to my original point: Given that "reflected-(reflected-me)" is (barring existence of additional mirrors) "original-me", we have that "reflected-x" makes sense regardless of the nature of "x". Thus, "reflected" --as "negative"-- describes an object's relationship to a counter-object, not its relationship to an intermediate object (the mirror, or zero).

That we refer to numbers less than zero as "the Negatives" is just a (ahem) reflection of our bias toward "the Positives", which seem --based on our foundational counting experience-- more like "actual" quantities.

I only use "minus" to describe the subtraction operation in arithmetic.

Edit to add:

Actually, I also use "minus" to describe a number less than zero: literally, a number that, via addition, makes a quantity smaller. (Latin "minus" = "less") Compare "plus": a number that, via addition, makes a quantity larger. (Latin "plus" = "more") Being a "plus" or a "minus" is an inherent property of the number's value, describing its relation to the intermediary, zero.

That said, I admit that I'm often a bit loose with my terminology. I might describe a number less than zero as "negative"; but, again, that's part of our bias toward what we call the positives.

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Then what a shame that the natural numbers have come to be called the "non-negative integers" most of the time. –  Marc van Leeuwen Sep 4 '13 at 11:49
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As far as I can check, in french this issue does not arise. The only available description is "moins x" or "moins 3", corresponding to "minus x" or "minus 3". The closest equivalent of "negative x" would be "x négatif" and it would only be used in phrases such as "pour x négatif", meaning "for x < 0" and not as a description of part of such a formula. "Négatif" seems to me mostly used in french in a categorical way "températures négatives", "nombres négatifs".

It would be informative if native german, russian, chinese, etc. speakers could comment on this.

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