# "Negative" versus "Minus"

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 $$-0.8$$?

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.

• @GEdgar: What is confusing about "minus zero point eight"? Jun 11, 2012 at 21:52
• 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, 2012 at 0:12
• $$\begin{array}{l}\text{We don't need no education}\cr\text{We don't need no thought control}\cr\text{No dark sarcasm in the classroom}\cr\text{Teacher leave them kids alone.}\end{array}$$ Jun 12, 2012 at 4:19
• Isn't this question better suited for English.SE rather than for Math.SE? Jun 12, 2012 at 4:45
• Why the $400$(!) bounty on this question... ? Jun 14, 2012 at 7:38

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.

• 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, 2012 at 1:47
• @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." Jun 18, 2012 at 4:46
• A small comment, but I do not consider operator overloading an abuse of notation. Sep 2, 2014 at 2:44

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"?

• @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? Jun 13, 2012 at 2:05
• @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? Jun 13, 2012 at 3:58
• Hmmmm ... I'm a mathematician, and I pronounce "-3" as "negative 3". :/
– Blue
Jun 14, 2012 at 7:02
• @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, 2014 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, 2014 at 16:26

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".

• 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. Jun 11, 2012 at 23:30
• @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. Jun 11, 2012 at 23:40
• Well, we certainly agree on times it by and matricee! Not to mention minus it by. Jun 11, 2012 at 23:55
• @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".) Jun 12, 2012 at 4:06
• Wow, why all the negativity? Minus 1 for all of you, including the answerer. Sep 7, 2012 at 14:01

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.

• 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. Jun 11, 2012 at 21:50
• 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. Jun 11, 2012 at 21:51
• 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 "$\rm -x$". Jun 11, 2012 at 23:19
• 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. Jun 12, 2012 at 4:08
• 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 5, 2016 at 19:42

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!

• There is nothing correct about your terminology. Nov 2, 2012 at 0:02
• 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. Nov 2, 2012 at 0:22

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

• As in (cf. answers below) Spanish, Danish, French... Is the distinction made in any other language? Jun 12, 2012 at 8:31
• Is this distinction made in British English? (I always say minus but I am 50 and English) Jun 12, 2012 at 9:50
• ... and in Portuguese. Jun 14, 2012 at 17:34
• ... and in Dutch (well, we use 'min'). Feb 12, 2015 at 16:56
• ... and in Hebrew. There are separate Hebrew words for "positive" and "negative", and the infix operators + and - have Hebrew words (although they are often pronounced as "plus" and "minus" instead). But the prefixes "+" and "-" are always pronounced "plus" and "minus". Jan 10, 2018 at 22:31

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.)

• @skullpatrol: Or ‘minus $5$’. And I don’t actually consider ‘the opposite of $5$’ correct: in my view opposite is not a technical term. Jun 12, 2012 at 21:25
• There is no real distinction between binary and unary minus: in the "unary" minus, it is simply $0$ that is left out. $-5$ means $0 - 5$. Aug 16, 2013 at 12:53
• 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. Aug 16, 2013 at 12:59
• P.S. Sometimes not only the language is adjusted to achieve goals, but the mathematics too. Aug 16, 2013 at 13:07
• 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. Aug 16, 2013 at 17:49

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.

• because if $s=-8$ then "negative s" is a positive number. Jul 9, 2015 at 19:41
• 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. Jul 9, 2015 at 20:47

"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.

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.

• Also in Argentina... always "menos 0.8", I've never heard "negativo 0.8" Jun 18, 2012 at 0:28

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.

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.

• 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. Jun 12, 2012 at 21:29

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...

• 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 $3$". Feb 19 at 17:38

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

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.

• In English, nothing is ever codified! Jun 12, 2012 at 4:15
• @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 :) Jun 12, 2012 at 4:17
• @Lubin: in fact, this makes me wonder if this question is not better suited for English.SE rather for Math.SE. Jun 12, 2012 at 4:27

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 .

• So you are comfortable with the assertion "negative $x$ is positive"? Jun 14, 2012 at 7:57

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".

• 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. Oct 16, 2014 at 12:15

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.

• 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, 2012 at 8:23
• I might. Yes, I might. Shall we, then, now commence upon a discussion of just how many nanoseconds suffice to clarify? :) Jun 14, 2012 at 19:17

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...

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.

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.

## 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.

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.

• 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.
– user856
Jun 14, 2012 at 17:33
• @RahulNarain fixed. Sorry for the inconsistency
– leo
Jun 14, 2012 at 17:40
• 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? Feb 19 at 17:37

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 $$5$$ and $$2$$. However, while scrolling through the responses, I became increasingly swayed in the minus direction.

The strongest arguments were

• the chorus of international observations that “negative $$3$$” is linguistically nonstandard (which supports that “minus $$3$$” is the more logical/fundamental cognitive construct while “negative $$3$$” is perhaps cultural);
• that $$\pm{3}$$ is always read as “plus minus $$3$$” anyway;
• and that the $$-$$ symbol when used as a prefix is generally more instructive (and less inconsistent) when construed as a (unary) arithmetic operator than as an adjective: $$7=-(-7)$$ “negative negative $$7$$”(?!) versus “minus minus $$7$$”.

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 3”; the character is opposed to the character meaning positive). (However, “minus $$3$$is colloquially used in some Mandarin-speaking regions.)

All things considered, I shall henceforth be saying “minus $$3$$” instead of “negative $$3$$”.

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

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

• Can you give some reference for this notation and terminology, please? I see it for the first time. Aug 17, 2013 at 9:28

In modern mathematics, there is not much of a difference between the unary minus and the binary minus anyway. '$$a-b$$' is most often regarded as an abbreviation of '$$a+(-b)$$', and so using different terminology seems superfluous.