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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.

I recently found an excellent historical/cultural perspective on What's so baffling about negative numbers? written by a Fields medalist.

Later
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user 85795
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    Avoid confusion. For $-0.8$ say "negative zero point eight" but for $-s$ say minus $s$. After a few years maybe allow minus in the first case also, but NEVER allow negative in the second case. – GEdgar Jun 11 '12 at 21:50
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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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    @GEdgar Why on earth "NEVER" allow "negative s"? That seems preferable than "minus s" to me, even at the lower levels. I presume you are trying to avoid the misperception that $-s$ is automatically a negative quantity, but I see the use of the word "negative" as a stronger counter to that misperception than the use of "minus". We have the negative of whatever $s$ is. – 2'5 9'2 Jun 11 '12 at 22:24
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    Minus was standard for me growing up... – copper.hat Jun 11 '12 at 23:15
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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 00:12
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    In my experience, in the USA, 6 - 0.8 is "6 minus 0.8" and -0.8 is "negative 0.8". Further, the only people I ever hear say "minus 0.8" for the latter are people who aren't very good at math (like someone in high school algebra while in college, I would not be surprised to hear this from them). I'm not saying this is how it has to be. I see Andre, who is definitely a better mathematician than I, does not see it this way. I'm just saying how it has been in my experience. – GeoffDS Jun 12 '12 at 00:12
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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 04:19
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    Isn't this question better suited for English.SE rather than for Math.SE? – haylem Jun 12 '12 at 04:45
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    The answers that argue against "negative" (I just read them all, so far) don't draw a distinction between the cases of a number or a variable. A number certainly can be negative, and a negative number is an entity in its own right. A negated variable like $-x$, on the other hand, is an expression which contains an irreducible operator. – Potatoswatter Jun 12 '12 at 04:55
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    Q: Does it really matter?
    A: Negative.     – Mathias Bynens Jun 12 '12 at 06:17
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    I've never heard anyone call minus negative, other than in the use of positive/negative charge. I suspect the use of negative is American. In English English we certainly use minus. – Nick Jun 12 '12 at 08:47
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    why did this quuestion get 23 upvotes? –  Jun 13 '12 at 09:45
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    I suggest we start denoting the unary function/additive inverse (depending on our point-of-view here) differently from the binary function. Maybe Nx for "negative" x, or @x, with "-" just for expressions (x-y). If we did that, and wrote our notes and books that way the equivocation that happens in something like (a+-b)=(a-b) wouldn't occur (how is "-" both unary and binary in the same equation? how does "-" signify both an additive inverse and a binary function?), since we'd write (a+Nb)=(a-b) instead. What we call the unary function then, I think, shouldn't much matter. – Doug Spoonwood Jun 14 '12 at 00:07
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    Why the $400$(!) bounty on this question... ? – Belgi Jun 14 '12 at 07:38
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    bounty because the question has not received enough attention? It has got 23 upvotes and 17 answers and more than 1K views.!! –  Jun 14 '12 at 08:35
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    The bounty is because skullpatrol, surely a charming young man, really, really, wants ammunition to aim at his instructor. I've never tried fixing notation, but I have taught weak students. Almost anything is worth a shot if it gives a chance of getting through. – Will Jagy Jun 14 '12 at 17:41
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    Shouldn't this question be closed as subjective and argumentative? –  Jun 15 '12 at 07:18
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    -1: This question is ridiculous--- "minus 3" or "negative 3" or "horizontal line to the left of 3" are all ways of speaking that produce a picture in your head, and all these debates are to produce a sense of superiority in some people because they "speak right" while other's don't. This is counterproductive to mathematical thinking, or to mathematical discourse. There are rare cases where speaking right is also thinking right, this isn't one of them. – Ron Maimon Nov 02 '12 at 00:03
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    Outside of the US, I believe standard English usage would be 'minus 8' rather than 'negative 8'. Fewer syllables and less attitude :-). – copper.hat Dec 29 '12 at 04:02
  • I am not a native English speaker, but to me "negative five" makes no sense: five is not negative, it is positive. – Alexey Aug 15 '13 at 17:31
  • @skullpatrol, in the phrase "serious joke", the adjective "serious" modifies the word "joke". This does not mean to me that the phrase makes sense (unless it is a joke). – Alexey Aug 16 '13 at 00:38
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    @skullpatrol, but you do not need to say "positive five" because there is only one five, and it is positive. There is no negative five. "Minus five" is a short way of saying "zero minus five". The opposite of minus five (-5 = 0 - 5) is five (5). – Alexey Aug 16 '13 at 08:28
  • Also, you do not say "imaginary one," or "imaginary five," but you say "imaginary unit", $i$, $5i$, etc. – Alexey Aug 16 '13 at 08:34
  • So you are asking that should we pronounce "minus" as "negative"? Well then, there is no reason to do so. "Minus" will remain our beloved "minus"... – user3459110 May 21 '14 at 13:03
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    Why is this question protected and not closed as off-topic? It could well fit to Mathematics Education site, but not here in my opinion. – ypercubeᵀᴹ Nov 04 '14 at 14:42
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    @alex.jordan — My eighth grade math teacher taught us to NEVER say "negative s" when referring to $-s$. Consider this: if $s=-8$, then what is "negative s"? It's positive 8. That's why we were taught to ALWAYS say "the opposite of s" and to never assume that $-s$ was a negative number. – chharvey Jul 09 '15 at 19:37
  • @chharvey "negative s" is "the negative of s", which in that example is "the negative of negative 8". To me with my background, "negative" already means "the opposite of", so it's a matter of background/culture/experience. (Also by the way, maybe "the opposite of 8" means "1/8", so there's no magic bullet.) But anyway, I think my comment (from three years ago!) was in response to GEdgar. I'm only making the point that the word "negative" is no better than the word "minus" when it comes to the potential for confusion here. – 2'5 9'2 Jul 09 '15 at 21:54
  • @alex.jordan Understandable, but put yourself in middle school students’ shoes… all their lives they’ve been told "negative" means "below zero." Also, btw, I believe in the world of mathematics education, the term "opposite" is explicitly reserved for "additive inverse" while the term "reciprocal" means "multiplicative inverse." – chharvey Jul 10 '15 at 02:02
  • There is certainly no harm in just using "minus" and never "negative", as this is the only way to say it in e.g. german, and it worked out fine for us. – M. Winter Nov 01 '20 at 09:55

26 Answers26

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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.

André Nicolas
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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). –  Jun 18 '12 at 01:47
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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 04:46
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    A small comment, but I do not consider operator overloading an abuse of notation. – goblin GONE Sep 02 '14 at 02:44
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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"?

Jonas Meyer
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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 02:05
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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 03:58
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    Hmmmm ... I'm a mathematician, and I pronounce "-3" as "negative 3". :/ – Blue Jun 14 '12 at 07:02
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    @AwalGarg: I wrote it two years ago, and I'll write it today: I pronounce "$-3$" as "negative 3". To elaborate: I consider "negative" to be unary; eg, "$-3$" describes a flavor of three. On the other hand, "minus" is binary: "$5-3$" is "five, minus three", indicating the subtraction of three from five. Blame influences of what Michael Hardy's answer terms "the imbecilic 'new math' of the late 60s", but the terminological nuances here make sense to me. So, I read "$2-8=-6$" as "$2$, minus $8$, equals negative $6$"; ie, "two, take-away eight, gives the number that [additively] cancels six". – Blue May 21 '14 at 14:20
  • @AwalGarg: I don't presume to call "minus" wrong, and I won't correct those who say it; heck, I say it myself from time to time. I simply prefer "negative", OK? (The seven up-votes on my comment suggest that I'm not completely alone here. :) As for "$+-3$": I'll admit "positive negative 3" sounds weird ... yet so does "plus minus 3"; but, since no one writes "$+-3$", the point is moot. :) That said, I read "$,^{-}(,^{-}3)$" as "negative negative three", but out-loud I'd likely say "the negative of negative-three". (BTW, I'm dis-inclined to debate opinions; expect no further replies.) – Blue May 21 '14 at 16:26
  • @AwalGarg: One more reply. Based on your subsequent answer, it appears that you intended to write "$\pm 3$" rather than the "$+-3$" that no one writes. (I thought that seemed odd, but since I assumed that you knew the TeX for "$\pm$", I figured you were making a point about weird symbological and terminological constructions. Be that as it may ...) As for "$\pm 3$": I read "$\pm 3$" as "positive-or-negative three"; but I read "$5 \pm 3$" as "five, plus-or-minus three". (Here, I lean toward saying that leaving out the "or" is wrong, but I accept it as a quirk of language. :) – Blue May 21 '14 at 16:45
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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".

rschwieb
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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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    I hate it when they say ... WARNING: LINK TO IMAGE – The Chaz 2.0 Jun 11 '12 at 23:49
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    @BrianM.Scott Declaration that you believe your opinion is the correct one acknowledged. – rschwieb Jun 11 '12 at 23:50
  • @TheChaz haha :P – rschwieb Jun 11 '12 at 23:51
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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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    @BrianM.Scott Agreed! – rschwieb Jun 11 '12 at 23:56
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    Matricee? What on earth does that mean, and how is it even read? – Bruno Stonek Jun 12 '12 at 01:22
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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 04:06
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    @JonasMeyer Probably no worse than matrixi ("matrix-eye") as a plural form. Although definitely weird. – Izkata Jun 12 '12 at 16:56
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    Wow, why all the negativity? Minus 1 for all of you, including the answerer. – Robusto Sep 07 '12 at 14:01
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    Haha! Now that you've written that, I suddenly see how the usage might be a linguistic preference. I prefer to describe the quantity with an adjective, while others prefer to just read off the symbol. Plus Positive 1 to your comment! – rschwieb Sep 07 '12 at 14:08
  • Are you sure they're not saying "matrices" (plural) – pshmath0 Sep 05 '13 at 07:31
  • @pbs From context, I'm absolutely certain they mean singular. They may say (or write) "this matrice," for example. – rschwieb Sep 05 '13 at 12:55
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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.

William
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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 04:08
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    Perhaps a better term than "negative $x$" would be "negated $x$". I agree with the possibility of confusion ("negative $x$" could be interpreted as "negative $\vert x \vert$"). – qman Oct 05 '16 at 19:42
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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!

Old John
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    There is nothing correct about your terminology. – Ron Maimon Nov 02 '12 at 00:02
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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 02 '12 at 00:22
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Like the answers above, I will also say that using "minus" in German is standard.

Cocopuffs
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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.)

Brian M. Scott
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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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    There is no real distinction between binary and unary minus: in the "unary" minus, it is simply $0$ that is left out. $-5$ means $0 - 5$. – Alexey Aug 16 '13 at 12:53
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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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    P.S. Sometimes not only the language is adjusted to achieve goals, but the mathematics too. – Alexey Aug 16 '13 at 13:07
  • @Alexey: On the contrary, there is a fundamental difference between a function that requires two arguments and one that takes only one argument, a difference that matters in such diverse fields as model theory and computer programming. – Brian M. Scott Aug 16 '13 at 13:11
  • @BrianM.Scott, i agree, but i am saying the meaning of $-x$, in my opinion, is a shorthand notation for $0-x$. This is how i always understood it, and this is a reason to not write something like $x - - y$. – Alexey Aug 16 '13 at 13:21
  • In my opinion, this is a natural way to extend to all integers the notation and operations that are first introduced only for positive integers. First a person learns the meaning of $5-4$, then $4-5$ and $-5+4$. It is best if all these expressions are parsed similarly. For that, the simplest way to do with the third one is to add the implicit $0$ in front (but of course after some practice "$-5$" looks like a number name, not "$-(5)$" or "$0-5$"). – Alexey Aug 16 '13 at 13:34
  • @Alexey: But $-x$ is not simply a shorthand notation for $0-x$, and there’s absolutely nothing wrong with $x--y$ apart from bad formatting: it should be $x-(-y)$ or, as in the old SMSG texts, $x-{}^-y$. ‘$-5$’ is a number name. – Brian M. Scott Aug 16 '13 at 13:38
  • @BrianM.Scott why is it not the shorthand notation? I mean, why can't it be viewed as such? And i do not think $x−−y$ is an acceptable form (i have only seen it in my students work, never in mathematics). Of course nothing is wrong with $x - (-y)$, as the parentheses are in place. I would read it $x - (0-y)$. – Alexey Aug 16 '13 at 13:42
  • @BrianM.Scott, unless you contradict yourself, your rules start looking complicated to me: you say that in "$-x$" the minus is a unary operation symbol, by in "$-5$" there is no operation symbol, it is a part of the number name. – Alexey Aug 16 '13 at 13:44
  • Well, of course you can view $\frac{1}{3}$ as a number name if you wish, for the lack of a better name, but first of all it is $1$ divided by $3$. – Alexey Aug 16 '13 at 13:49
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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
  • @Alexey: I, on the other hand, am familiar with the SMSG texts at first hand: back in the 1960s I owned a complete set, including the teacher’s manuals. They were actually rather good. – Brian M. Scott Aug 16 '13 at 17:56
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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
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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.

Lubin
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    because if $s=-8$ then "negative s" is a positive number. – chharvey Jul 09 '15 at 19:41
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    Of course, @chharvey, and that’s exactly how I would justify my position if I followed my preferences. But if I’m going to kick against a bad practice of modern teaching, I’ll reserve my kicks for truly damaging pedagogical practices. – Lubin Jul 09 '15 at 20:47
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"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.

Michael Hardy
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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.

Bouvet Island
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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.

utdiscant
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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.

Mark Bennet
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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
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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...

user3459110
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    I disagree. In the quadratic formula (for example), $\pm$ is pronounced "plus or minus" (it is being used as a binary operator here), but I would read the bare notation $\pm 3$ as "positive or negative $3$". – Xander Henderson Feb 19 '22 at 17:38
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What a fuss about nothing! It's like "math" versus "maths" -- that is to say, simply a question of local convention.

TonyK
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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.

haylem
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    In English, nothing is ever codified! – Lubin Jun 12 '12 at 04:15
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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 04:17
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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 04:27
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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.

David Wheeler
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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. –  Jun 14 '12 at 08:23
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    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
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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 .

tronz
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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".

IAmNaN
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    As this the only mention of CS here, I'd like to expand upon it. There are computer languages, where negation is not expressed by "minus" symbol! J is one example, Calc is another one. In both the way to write negative numbers is to put an underscore before the digits. Obviously, there's also a binary representation, which writes numbers in an even more different way. So, even if historically both forms used to mean the same thing, to make it future-proof, it's better to distinguish between the two. – wvxvw Oct 16 '14 at 12:15
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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...

Sniper Clown
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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.

ncmathsadist
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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.

BLAZE
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MaTECmatica
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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.

Martin Thoma
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I prefer this convention:

  • Positive 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.

jvdhooft
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leo
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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. –  Jun 14 '12 at 17:33
  • @RahulNarain fixed. Sorry for the inconsistency – leo Jun 14 '12 at 17:40
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    Just piping up to voice disagreement with user856: we can define words to mean whatever we want them to mean. The standard definitions of positive and negative are such that $0$ is neither. However, I often find it useful to define these terms such $0$ is both positive and negative. As long as my usage is clear and consistent, who cares? – Xander Henderson Feb 19 '22 at 17:37
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I had chanced upon this page with nary a doubt that ‘negative $3$’ is the superior choice (and ‘minus $3$’ mildly puerile) to express the negative difference between $2$ and $5$. But after digesting the various responses here, I've changed my position: ‘minus $3$’ is the only correct reading, because:

  1. the chorus of international observations that ‘negative $3$’ is linguistically nonstandard suggests that ‘minus $3$’ is the more logical/fundamental cognitive construct while ‘negative $3$’ is perhaps cultural.

  2. ±3 is always read as ‘plus (or) minus $3$’ anyway.

  3. the - symbol when used as a prefix is as a unary arithmetic operator, and makes sense neither as a specification of a negative number (after all, -(-7) is certainly positive!) nor as the adjective ‘negative’ (after all, the - in -5 is certainly neither indicating that $5$ is negative nor that $5$ has both a positive and a negative version!). Calling -(-7)$=7\,$ ‘minus minus $7$’ is perfectly coherent, whereas ‘negative negative $7$’ is actually pretty illogical.

    Another example: when $b$ is an unknown constant, reading -b as ‘minus $b$’ neither assumes that -b is negative, nor assumes that $b$ is negative; on the other hand, does the reading ‘negative $b$’ mean that $b$ is negative or that -b is negative (each interpretation makes an unwarranted assumption)?

P.S. Interestingly though, it appears that Mandarin Chinese runs counter to the European languages in reading $$2-5=(-3)$$ as ‘$2$ minus $5$ equals negative $3$’ (‘$2$ 减 $5$ 得 负 $3$’; the character 负 is opposed to the character 正 meaning positive). However, some Mandarin-speaking regions colloquially do say ‘minus $3$’.

ryang
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It is common in mathematics to use the same word to describe distinct phenomena if they happen to be closely linked. For instance, in abstract algebra, the word "order" is used in relation to groups, and also to elements of a group. With this in mind, pronouncing $a-b$ as "$a$ minus $b$" makes a lot of sense, given that by definition, $a-b=a+(-b)$; that is, the definition of the binary minus involves the definition of the unary minus.

Joe
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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\} $

Elements In Space
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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 09:28