How should these be read? $-x$ , $-(2)$ , $-2x$ , $\pm x$

Question

The minus sign represents a number of concepts in maths.

The basic two are negative numbers and subtraction.

For example: $2-(-2)$ , would be read “two minus negative two”.

When teaching maths to middle school children I get them to make this distinction when reading. It forces then to distinguish the concepts of negative numbers and the subtraction operation. Sal Khan from Khan Academy always makes this distinction. He would never read -2 as minus two. It is a negative two.

I agree that saying ''minus'' every time is common even among mathematicians. But they already understand the function of the sign in the context that they see it. So $-(-3-(-1))$ could be read as ''minus minus three minus minus one''. But I would like students and myself to be able to distinguish what the sign is doing in each case and express that in words. ''the additive inverse of negative three minus negative one''.

But what about $-x$?

We don't know if $x$ is positive or negative and this is not subtraction. I read somewhere that this should be read as “the additive inverse of $x$”. Is this true ?

If $-x$ would be read as “the additive inverse of x” then what about $-(2)$ ? Would it be ''the additive inverse of two''.

Thirdly, what about a number-variable term like $-2x$ ? “the additive inverse of two x”?

Finally, if $-x$ is “the additive inverse of $x$” then what about $\pm x$ ? ''the something or inverse of $x$”. What is the opposite of additive inverse ?

Just say "negative $x$", "plus or minus $x$," etc., keep it simple. — Qiaochu Yuan, Aug 21 '20 at 09:36
Indeed, what do you mean by "the opposite of additive inverse"? The inverse of the inverse is the element itself, i.e., $-(-x)=x$. So I would avoid the word "opposite", since this also has other meanings in mathematics. — Dietrich Burde, Aug 21 '20 at 09:38
@DietrichBurde Take some $x$. We don't know whether $x$ is positive or negative. Now take $y=\pm x$. We can see that $y$ has two values, one of them is the additive inverse of x, the other is just x. But is there a way of expressing ''just x as opposed to the additive inverse of x'' ? — Kantura, Aug 21 '20 at 09:44
@AnginaSeng Minus means to subtract. There is no subtracting (minusing) happening in any of those expressions. — Kantura, Aug 21 '20 at 09:47
What do you mean by "opposed to" or "opposite"? We just have $y=x$ or $y=-x$, right? — Dietrich Burde, Aug 21 '20 at 09:49
@DietrichBurde I mean ''don't take the additive inverse''. Is there a term in English for this ? If $-x$ is the additive inverse of $x$ then how would you read $\pm x$ ? — Kantura, Aug 21 '20 at 09:58
@AnginaSeng Yes there is. ''to minus'' is a verb which means ''to subtract''. ''minusing'' means ''subtracting''. It is the present participle of minus. Here are some links. https://en.wiktionary.org/wiki/minusing https://en.wiktionary.org/wiki/minus#Verb — Kantura, Aug 21 '20 at 10:16
I am not authoritative, and I always say minus. $2-(-x)$ is two minus minus $x$. — , Aug 21 '20 at 10:18
@Kantura I have added those "who use minus as a verb" to my little list. — Angina Seng, Aug 21 '20 at 10:21
I take exception to your writing "the minus sign ... represents negative numbers." In the expression $-(-2)$, the first minus sign represents neither subtraction nor a negative number; both minus signs represent additive inverse. You don't have to read it as additive inverse, if you don't want to; it's perfectly OK to read it as minus minus two. But you have to keep in mind that you're talking about the additive inverse. That way, you won't be confused into thinking $-x$ is a negative number. It's negative if $x>0$, it's positive if $x<0$, but in all cases it's the additive inverse of $x$. — Gerry Myerson, Aug 21 '20 at 10:56
@GerryMyerson Hmm ... just to clarify, do you take exception with the statement ''The minus sign is used among other things to represent the notion of negative.'' ? — Kantura, Aug 21 '20 at 11:15
I worry that that statement could be misleading. Sure, the minus sign in $-2$ distinguishes the negative number $-2$ from the positive number $2$, but I've seen too many students assume that $-x$ has to be a negative number to be comfortable with the statement you propose. — Gerry Myerson, Aug 21 '20 at 11:20
@GerryMyerson That statement is not mine, it is a quote from Wiki. I agree with you that students often make false assumptions regarding the minus sign. That's why one of the first things I teach them is the different concepts that the minus sign represents. If they understand the difference between negative, subtraction and inverse, they are less likely to make a minus sign related mistake in future. Distinguishing these three concepts in language forces them to distinguish what the sign is doing in each case. — Kantura, Aug 21 '20 at 11:30
This might be a better fit at https://matheducators.stackexchange.com — Barry Cipra, Aug 21 '20 at 11:58

score 2 · Answer 1 · answered Aug 21 '20 at 10:57

IIRC, Al-Khwarizmi, in his book where he defined "al-jabr" (completion, rejoining, balancing; the origin of the term "algebra"), describes a negative quantity as a "debt" (his book is mostly text, as symbolic mathematics weren't even close to what they are today). You can illustrate the concepts of debt and credit with basic exercises of "who owes a certain number of candy to whom".

As for "reading out loud with words", I think it might not be the best approach, because it quickly gets confusing, and I'd encourage your students to think visually, because it's a lot easier. Present the integers as a ladder that is infinite on both sides: negative numbers make you take a certain number of steps down, positive numbers a certain number of steps up.

This is additionally a very effective perspective to have when moving on to linear algebra, as the "ladder" image is basically understanding real numbers (a fortiori integers) as vectors in $\mathbb{R}^1$.

As for the algebraic aspect of subtraction, I've found that the idea that "subtraction is addition by the opposite" (which you mention) and that "division is multiplication by the inverse" is considerably useful. It comes from the very axioms of "groups", in the sense of algebraic structures. In a nutshell, teach your students that if they're confused with their symbols, they can do:

$$a - b = a + (-b)$$

$$a / b = a * (1/b) $$

And that often helps two minuses or two divisions cancel out, while bringing things back to operators which are a lot nicer to handle (addition and multiplication are both associative and commutative, while subtraction and division are not). You can also present the "two minuses cancel out" idea by saying that "a minus is the opposite of the direction of your steps, so if you go the opposite of 3 steps down, then you're going three steps up". The preceding formula $a - b = a + (-b)$ can similarly be explained with this idea that "the opposite of going three steps up is going three steps down".

Hope this helps.

score 0 · Answer 2 · answered Jun 15 '23 at 14:39

How should these be read? $-x$

“Minus $x$”.

Here, "minus" is a negation (operation), not an adjective, and “minus $x$” need not be negative.

Minus means to subtract. There is no subtracting (minusing) happening in any of those expressions.

There is implicit subtraction in the expression −x, which means $0-x.$

Alternatively, think of the - in -x as a unary operator.

$-2$

“Minus two”. This is also literally how many European languages read it.

$2-(-2)$ , would be read “two minus negative two”.

When teaching maths to middle school children I get them to make this distinction. It forces then to distinguish the concepts of negative numbers and the subtraction operation.

Reading 2-(-x) as “two minus negative $x$” is potentially confusing or even misleading: what if $x$ is $-7,$ or if $x$ is $7+2i$ ?

Better to say “two minus minus $x$”, especially if the goal is to be instructive or systematic.

Now take $y=±x.$ We can see that $y$ has two values

No, the placeholder $y$ represents just one value: either $x$ or $-x.$

$\pm x$

“Plus (or) minus $x$”.

z=±x is not a genuine equation, but instead a disjunction: it means that $z=x$ or $z=-x.$

How should these be read? $-x$ , $-(2)$ , $-2x$ , $\pm x$

2 Answers2