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As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$?

The so called "textbook answer" regarding this question reads:

A number and its opposite are called additive inverses of each other because their sum is zero, the identity element for addition. Thus, the numeral $-5$ can be read "negative five," "the opposite of five," or "the additive inverse of five."

This question involves two separate, but related issues; the first is discussed at an elementary level here. While the second, and more advanced, issue is discussed here. I also found this concerning use in elementary education.

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@GEdgar: What is confusing about "minus zero point eight"? –  Chris Eagle Jun 11 '12 at 21:52
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As a by-the-way, negative eight was unheard of in my schooling in Ireland. I think it's an American phenomenon. –  TRiG Jun 12 '12 at 0:12
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$$\begin{array}{l}\text{We don't need no education}\cr\text{We don't need no thought control}\cr\text{No dark sarcasm in the classroom}\cr\text{Teacher leave them kids alone.}\end{array}$$ –  Will Jagy Jun 12 '12 at 4:19
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Isn't this question better suited for English.SE rather than for Math.SE? –  haylem Jun 12 '12 at 4:45
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Why the $400$(!) bounty on this question... ? –  Belgi Jun 14 '12 at 7:38

25 Answers 25

up vote 16 down vote accepted

As a retired teacher, I can say that I tried very hard for many years to get my students to use the term "negative" instead of "minus", but after so many years of trying, I was finally happy if they could understand the concept, and stopped worrying so much about whether they used the correct terminology!

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There is nothing correct about your terminology. –  Ron Maimon Nov 2 '12 at 0:02
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That depends totally on who is writing the rules for what is "correct". At the time "negative was certainly "the correct terminology". Mathematical terminology has changed over the years, and although we may say that newer terminology is preferable in some way, it is meaningless to say that one is correct and another is incorrect. –  Old John Nov 2 '12 at 0:22
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+1 Well put, read "plus one." –  skullpatrol Jul 11 '13 at 6:12

I am fully comfortable with "minus $x$," and indeed like it better than "negative $x$," and have seldom used the latter in lectures.

There is no problem with the binary operator and the unary operator having the same name. Speaking and writing mathematics would be more awkward if we did not allow useful abus de langage.

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Umm...shouldn't that be 'langage'? –  TonyK Jun 17 '12 at 23:34
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There is a substantial difference between $-x$ and $-5$. "Negative x" is likely to create confusion: do you mean $-x$, or that $x$ is negative? Not to mention that $-x$ may well be zero or a positive number. Anyway, the OP did not ask us about "negative x" vs "minus x"; the question was about "negative 0.8" vs "minus 0.8".So this answer does not address the question. Hence, -1. (Read as "minus one", but that's just my preference). –  user31373 Jun 18 '12 at 1:47
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@LeonidKovalev: Here $x$ was a placeholder. It is intended that the the comment applies in more or less uniformly, so for example my preferred pronunciation of "-5" is "minus five." There are exceptions. When one refers to temperature, one may say "five below zero." –  André Nicolas Jun 18 '12 at 4:46

From page 271 of Halmos's I want to be a mathematician:

Here is a bit of innocent fun that is not much of a challenge, but most calculus students seem to enjoy it. Partly as integration drill and partly to make a point about the use of "dummy variables", I'd call on several students, one after another, and demand that they tell me what is $\displaystyle\int\dfrac{dx}{x}$, $\displaystyle\int\dfrac{du}{u}$, $\displaystyle\int\dfrac{dz}{z}$, $\displaystyle\int\dfrac{da}{a}$, and then, as the clincher, I'd ask about $\displaystyle\int\dfrac{d(\text{cabin})}{\text{cabin}}$. Some of them would grin amiably and shout out "log cabin", and they were surprised when I told them that I didn't agree. The right answer (as I learned when I was learning calculus) is "house-boat", "log cabin plus sea".

At the same time, by the way, I'd take advantage of the occasion and tell my students that the exponential that $2$ is the logarithm of is not $10^2$ but $e^2$; that's how mathematicians use the language. The use of $\ln$ is a textbook vulgarization. Did you ever hear a mathematician speak of the Riemann surface of $\ln z$? And speaking of vulgarizations, did you ever hear a mathematician pronounce "$-3$" as "negative three"?

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A number and its opposite are called additive inverses of each other because their sum is zero, the identity element for addition. Thus, the numeral −5 can be read "negative five," "the opposite of five," or "the additive inverse of five." –  skullpatrol Jun 12 '12 at 20:50
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@skullpatrol: That is the same thing you added to your question claiming it is "the textbook answer." Is it a quote from something? If so, will you please give the source? Why are you adding this comment to numerous answers when you already put it in the question? –  Jonas Meyer Jun 13 '12 at 2:05
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@skullpatrol: I agree with Jonas here - I don't understand why you're posting this same comment on so many answers. In fact, your comments have been flagged by users as spam, and I'm inclined to remove them. Why wasn't the addition to the question enough? –  Zev Chonoles Jun 13 '12 at 3:58
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Hmmmm ... I'm a mathematician, and I pronounce "-3" as "negative 3". :/ –  Blue Jun 14 '12 at 7:02
    
@Blue So you pronounce minus 3 as negative 3? –  Awal Garg May 21 at 13:06

I would encourage (maybe insist is too strong) to use "negative". It's not the worst idiosyncrasy, though. I prefer this distinction so that the unary "-" and binary "-" are two different things.

It irritates me a little more when students say "times-ing it by 5", or "matricee".

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That is presumably why the negative usage was introduced, but over the years I’ve seen little evidence that it’s had the intended effect. I strongly suspect that it just adds an extra opportunity for confusion, so I’ve never seen any reason to change my usage from minus 3 to negative 3. My students have already been thoroughly indoctrinated in the use of negative 3 and need no encouragement. –  Brian M. Scott Jun 11 '12 at 23:30
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@BrianM.Scott I think people are interpreting my post a little too strongly: my feelings are more 55%-45% split. It would certainly be impractical to "insist". I would worry about teachers getting into the habit of sweeping differences like this under the rug. As an example of something I feel is very similar, I heard of a college teacher telling students not to bother with "$dx$" in integrals. This led to bad performance in integration by substitution and by parts. –  rschwieb Jun 11 '12 at 23:40
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But I don’t think that the two are at all similar. I think that the terminological distinction is rather pointless and would encourage people not to teach it; the $dx$ in the integral is another matter altogether. –  Brian M. Scott Jun 11 '12 at 23:44
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Well, we certainly agree on times it by and matricee! Not to mention minus it by. –  Brian M. Scott Jun 11 '12 at 23:55
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@Bruno: Students hear people say "matrices", then when they refer to a single matrix, they try to back-form the singular by dropping the "s". Similarly, the singular of "vertices" becomes "verticee". (The double "e" at the end is used to indicate how it sounds to English speakers, rhyming with "glee".) –  Jonas Meyer Jun 12 '12 at 4:06

I don't understand why you would encourage using "negative". The term "negative" has meaning only in structures that have an ordering.

More generally and often the property of $-a$ that one uses is that fact that $a + (-a) = 0$, i.e. $-a$ is the additive inverse. In this case, it should be read minus $a$, and definitely not negative $a$ if one is in a situation where the structure does not have an ordering.

I would encourage using "minus" $a$ since "minus" and "negative" $a$ agree in ordered rings while "negative" is not correct in an algebraic structure without order.

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Even for real numbers, the reading of $-x$ as "negative $x$" may tend to increase confusion, because if $x$ happens to be negative, $-x$ is positive and not negative. –  Robert Israel Jun 11 '12 at 21:50
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The reasoning is easy: the unary "-" and binary "-" are completely different operations, conceptually. Therefore one could insist on different names for different things. I don't understand why order is so important... but I understand that it is the (different) viewpoint you have. –  rschwieb Jun 11 '12 at 21:51
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It's not necessarily true that use of "negative" implies a linear ordering. For example, many authors write "negate x" for "invert x" in commutative groups. In such contexts "negative x" means "$\rm -x$". –  Bill Dubuque Jun 11 '12 at 23:19
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In agreement with @Bill, I’ve always assumed that “negative 8” meant “the negative of 8”, in other words the additive inverse of 8. But confusion will still arise (perhaps is even more likely to arise) when $-s$ is positive. –  Lubin Jun 12 '12 at 4:08

Like the answers above, I will also say that using "minus" in German is standard.

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As in (cf. answers below) Spanish, Danish, French... Is the distinction made in any other language? –  D. Thomine Jun 12 '12 at 8:31
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Is this distinction made in British English? (I always say minus but I am 50 and English) –  Mark Jun 12 '12 at 9:50
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... and in Portuguese. –  Américo Tavares Jun 14 '12 at 17:34

Absolutely not. The introduction of this use of negative was well-intentioned but did little or nothing to improve students’ understanding of the distinction between binary and unary minus. Those students who understand that there’s a difference between unary and binary minus don’t really need a terminological distinction, and for those who don’t it’s just a potential additional source of confusion. I continue to say minus 3, as I always have done. (Mind you, either a lot of high school teachers are insisting on negative 3, or, more likely, that usage has simply become a largely unquestioned standard, because virtually all of my students for a good many years now have automatically said negative 3.)

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@skullpatrol: Or ‘minus $5$’. And I don’t actually consider ‘the opposite of $5$’ correct: in my view opposite is not a technical term. –  Brian M. Scott Jun 12 '12 at 21:25
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In your view Sir, is "the additive inverse of five" a technical phrase? –  skullpatrol Nov 1 '12 at 22:51
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It is incredible in my opinion how educators can succeed in turning even mathematics upside down. Instead of teaching mathematics how it is, they seem to make "intentions," set "goals," introduce "methods," and try to adjust mathematical language to achieving these goals with these methods, with best intentions. –  Alexey Aug 16 '13 at 12:59
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I have just read in Wikipedia what SMSG was, and that it was created in the wake of "Sputnik crisis." I think that the reason for the Sputnik crisis was probably that US Americans were writhing $x - {}^-y$ :D (lol). P.S. If anyone finds this comment offensive, i will apologize and remove it. –  Alexey Aug 16 '13 at 17:49
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@BrianM.Scott, maybe they are good, i trust you, i indeed do not know them. My "funny" comment is not serious, i may remove it. The reason for my extended comments was that when i was a TA in the US, and student were saying "negative two," i felt like saying: "Wait, just think what you've just said, what is 'negative 2'?" But English was not my native language, and i decided that maybe it was the standard English usage, maybe coming from the times of Shakespeare. And here today i've learned that this usage was deliberately introduced by educators! (As a roundabout way to make math easy?) –  Alexey Aug 16 '13 at 18:05

I’m old enough that I can remember a time when one never said “negative 8” for $-8$; and I’m so old that I can’t recall just when the newer usage became current. But in working with high-school students these days, I try to say “negative 8” so as not to confuse them. I really like the injunction to never say “negative $s$” for $-s$, but I think I’d have trouble convincing them why, when asked to explain.

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It's strange, in spanish (my mother language) we tend to say "menos 0.8" instead of "negativo 0.8" (I think no translation is needed, right?)

So it seems that the concept is more important than how we say it.

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Also in Argentina... always "menos 0.8", I've never heard "negativo 0.8" –  leonbloy Jun 18 '12 at 0:28

"Minus 3" used to be the standard way to read "$-3$". I think "negative 3" was introduced along with the imbecilic "new math" of the late '60s. Prior to that, one used the word "negative" only in such expressions as "The product of two negative numbers is positive" and "Both solutions of this equation are negative".

This is one of the usages that Paul Halmos ridiculed in his autobiography, saying mathematicians didn't use the term and teachers shouldn't be teaching it.

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In Danish, the more correct term is actually "Minus 0.8" and not "Negativ 0.8". Personally, this is also what I prefer in English.

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It seems to me that there are two aspects to this question.

One is clarity of mathematical thought, and there may be contexts in which "negative" is more precise than "minus" in this context.

Another is teaching students to communicate effectively with each other and to understand their text books - I would say that, at the elementary level at which negative numbers are first encountered, "minus" is standard language: to teach students in this context that "minus" is wrong and "negative" is right would seem to me more likely to impede communication than to enhance it.

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It can be read as all those things, of course. Amongst the people I know it is normally read as "minus 5" and is a negative number. So if I want to communicate easily and efficiently to the people I know, I say "minus 5" because that is what they expect and understand. It isn't a question of mathematics so much as the way we use language to communicate. –  Mark Bennet Jun 12 '12 at 21:29

Maybe it's because I'm not English-native (or, in my referential, maybe a lot of people do the same mistake in French and German), but "minus" is standard from what I know in these languages.

Plus you can refer to a number as being negative, but any variable could hold a negative value already, and reading it "negative X" would in my sense strongly influence the thought-process about X.

But:

  • I'm not a mathematician,
  • I'm not a member of the French Academy or a grammarian to decide this.

Still I'd assume this has been codified somewhere for my language and for English as well.

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In English, nothing is ever codified! –  Lubin Jun 12 '12 at 4:15
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@Lubin: yeah, the French Academy is both a blessing and a curse. Blessing as it takes out the doubt, curse as you're very often wrong about what the correct usage should be and you feel easily alienated by your language :) –  haylem Jun 12 '12 at 4:17
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@Lubin: in fact, this makes me wonder if this question is not better suited for English.SE rather for Math.SE. –  haylem Jun 12 '12 at 4:27

What a fuss about nothing! It's like "math" versus "maths" -- that is to say, simply a question of local convention.

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Negative is more appropriate than minus if it comes to denote the negative term like -0.8 . While minus is used as a binary operator like (a-b) a minus b .

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So you are comfortable with the assertion "negative $x$ is positive"? –  Martin Argerami Jun 14 '12 at 7:57

Before moving to USA, I was educated in the British system, where minus x was more prevalent than negative x. I also had to adjust to radical x and distinguish parenthesis from brackets. Although, in hindsight it was frustrating and having a convention would have made my life easier, certain bit of asymmetry is necessary for the beauty echoing André Nicolas's response.

For instance, even though the following should be the strict convention as it would highlight the pattern easily to the uninitiated and young children:

$$ \frac{1}{1} + \frac{1}{2} + \frac{1}{3}$$

we prefer the asymmetrical:

$$ 1 + \frac{1}{2} + \frac{1}{3}$$

because we assume certain intelligence in mathematics and part of a student's curriculum should be how to code-switch from different notations.

Also, a point worth remembering before reinventing the wheel, seminar involving mathematicians will take place for to debate and if a formal convention is adopted, it would involve costs to change the books et al.

Really it's a matter of cracking an either side of egg...

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I have almost always said, "minus." What is interesting here is that the - operator has two guises. It is an infix binary operator (as in $5 - 3$) and it is a prefix unary operator, as in $-7$.

The word "negative" has the liability of an extra syllable. Occasionally, I do find myself saying "negative 3" though.

This seems to me to be a distinction without a huge difference.

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I'm not sure what's at stake, here, or what question is actually being asked.

In my own mind, I tend to use "minus 5" and "negative 5" interchangeably, and it seems that the shift in usage from the former to the latter is primarily an example of the malleability of language over time. I am 50 years old, and I have seen one usage become "old-fashioned."

There is, however, one instance in which "negative" versus "minus" is clearly superior:

If I say: "Nine, negative five" it is clear I am enumerating two numbers: $9$, $-5$. If I say: "Nine minus five," it is unclear whether I intend $9 - 5$ (that is: $4$), or the list $9$, $-5$.

Historically speaking (and this history is mirrored somewhat in language), subtraction predates the creation of integers. "Minus" comes from the Latin word for "less", and its usage in subtraction reflects this origin. "Negative" comes from the Latin verb "to deny" (and most likely, by extension, to cancel), implying a more sophisticated social structure than our early beginnings.

As mathematical systems have becomes more abstract, it seems logical to me that "negative" is the term more usefully applied to things such as elements of an abelian group (where the operation "+" may bear little resemblance to "adding things"). For example, I would not call the matrix $-A$, "minus A". But that's just "my" personal take on things, and I do not claim to speak for the community at large in any substantial fashion.

I fail to see the point of belaboring terminology, you could call negative numbers "floompsies", as long as you correctly capture their behavior.

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You might say "nine [long pause] minus five", or you might say "nine minus five" without much of a pause. The distinction would be clear. –  user22805 Jun 14 '12 at 8:23
    
I might. Yes, I might. Shall we, then, now commence upon a discussion of just how many nanoseconds suffice to clarify? :) –  David Wheeler Jun 14 '12 at 19:17

I was a teacher of computer science, not math. I preferred 'negative' when lecturing. However, the zero is implied and therefore correct.

Further, it's a slippery slope. You would also need to insist they use the same vernacular when describing measurements, as in "minus 10 degrees".

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A practical situation where the difference between unary (as in negative 0.8) and binary (as in 1.0 minus 0.8) is important is when using Microsoft Excel. For this spreadsheet program, the unary and binary operators have a different hierarchy, therefore if you enter:
=10-4^2
in an Excel cell, the answer you get is -6, however if you enter:
=-4^2+10
the answer you get is different, it is 26. Other computer programs do Not behave that way, for example, if you use Mathematica and you enter:
10-4^2
and
-4^2+10
in both cases you get -6, because unlike Excel, Mathematica has the same hierarchy for both the unary and binary -. I find this issue (the behavior of Excel different from the common behavior of other software) very important to teach to my Engineering students.

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I prefer this convention:

  • Positeve number: if the number is strictly greater than $0$.
  • Negative number: if the number is strictly less than $0$.
  • $0$: $0$ is not positive nor negative.

Then $-x$, "minus $x$", and "negative $x$" are just what they are. Particularly if $x$ is negative, minus $x$ is positive. I interpret "negative $x$" as $x$ a negative number. Minus $x$ as $-x$ and it depends on $x$ if minus $x$ is positive or negative.

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You don't get to decide that zero is both positive and negative. "Positive" means strictly greater than zero, and "negative" means strictly less than zero. Zero is not strictly greater or less than zero. –  Rahul Jun 14 '12 at 17:33
    
@RahulNarain fixed. Sorry for the inconsistency –  leo Jun 14 '12 at 17:40

How to teach the difference

I think you should give your students $(\mathbb{Z}, -)$ and $(\mathbb{Z}, +)$ as an example and let them check both objects for

  • associativity
  • commutativity
  • neutral element (left neutral / right neutral)
  • inverse elements

I am a computer science / math student and this was multiple times part of assignments:

  • Check if $(\mathbb{Z}, -)$ and/or $(\mathbb{Z}, +)$ are groups. Proof or find all reasons why not.
  • Find a set and an operation that is a magma, but not a semigroup
  • Find a set that is as small as possible that generates $(\mathbb{Z}, -)$. Do the same for $(\mathbb{Z}, +)$.

Language

I come from Germany and there is no such distinction by language. You always say "minus 0.8".

However, we do know the word "negativ". When you say a number is negative, you mean it is smaller than $0$. I think it's the same in English.
But the word "negative" is never used like "negative 0.8". It's used like

Minus 0.8 is a negative number.

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Although this is old, I would like to add a point missed by many people...

"Minus" corresponds to the correct word/terminology.

Proof

$\pm{3}$ is pronounced as "Plus Minus three" in any case. It is not pronounced "positive negative three" by anyone, and I think everybody here would agree with this.

And, by hindsight, one can argue that the "plus" is for the symbol $+$ and the "minus" is for the symbol $-$.

Therefore, "minus 3" is the correct terminology for $-3$.

No need to clap...

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All three have issues.

  1. As you point out, the terminology "minus 3" is technically incorrect.

  2. As others have pointed out, the terminology "negative 3" is misleading, because "negative $x$" needn't be negative, e.g. take $x=-3$, then $-x$ is positive.

  3. Saying "the additive inverse of 3" is far too long winded.

What comes next depends on your philosophy of teaching.

If your philosophy is, "well its too late to change the terminology, I have to give these kids the best education I can, given the constraints," then probably don't worry too much about it, but do make sure to emphasize that negative $x$ is the reflection of $x$ across the point $0$, and therefore needn't be negative. You should also emphasize that plus has nothing to do with positive numbers. E.g. the positive numbers are closed under addition, and so too are the negatives. Indeed, the asymmetry between positive and negative only emerges once multiplication is introduced.

On the other hand, if your philosophy is more like, "NO!! Its not too late. I AM A HUMAN BEING DAMNIT. MY LIFE HAS VALUE." then you should probably come up with your own terminology. And hey, you might just start something. Some ideas for terminology:

$-x = $ reflect $x$, reflection $x$, flip $x$, reversal $x$, antipose $x$.

That last one is motivated by the fact that, in the complex plane, we can always draw a unique circle centered at $0$ passing through $x$, in which case antipose $x$ is the unique complex number diametrically opposed.

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How about the opposite of x? –  skullpatrol Sep 4 '13 at 20:10
    
@skullpatrol, yeah that could work. However, you need to contract it. "$-x$ = opposite $x$," not "$x$ = the opposite of $x$." Otherwise, it simply won't catch on. Another option, which I failed to mention in my answer, is to keep reading $-x$ as "negative $x$," but to rename the two halves of the real line. Maybe have a "negital" and a "posital" half. –  goblin Sep 5 '13 at 7:32
    
@skullpatrol, personally like "flip $x$" the best. Your students have a preference for "minus" over "negative" because it has fewer syllables. Well, "flip" has one less syllable again! –  goblin Sep 5 '13 at 7:34
    
Well, if you want to be as brief as possible how about: $-a$ = "op" a. –  skullpatrol Sep 5 '13 at 9:22
    
@skullpatrol, that could definitely work. Along a similar vein, how about: $-a =$ "neg" $a$. That way, your students will still be comprehensible to the other students whom haven't received your awesome teaching! –  goblin Sep 5 '13 at 10:31

In logic

the negation of a certain element in a set is all the other terms in the set

for the set $\{1,2,3,4\}$

the negation of the element 2 is

$\neg 2=\{1,3,4\} $

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Can you give some reference for this notation and terminology, please? I see it for the first time. –  Alexey Aug 17 '13 at 9:28

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