"Negative" versus "Minus"

Question

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.

Answer 1

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".

Answer 2

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!

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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

Answer 9

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.

Answer 10

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.)

Answer 11

"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.

Answer 12

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"?

Answer 13

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.

Answer 14

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 .

Answer 15

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".

Answer 16

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.

Answer 17

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.

Answer 18

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.

Answer 19

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...

Answer 20

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

Answer 21

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.

Answer 22

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\} $

Answer 23

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 $3$’.

Answer 24

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.