# Parentheses around negative numbers

We teach students that a notation like $$17 - -59$$ is not acceptable or at least not good. Instead we want them to write $$17-(-59)$$ The main reason seems to be that it's more readable if you separate the operator and the number sign.

But since

• students frequently make errors regarding this notation,
• the notation is not really needed to avoid ambiguity and
• there is that weird break of symmetry since only use brackets for the subtrahend and not the minuend

I wonder to just accept the version without parentheses or even use it myself in class.

Are there any reasons why this notation is necessary?

• Probably not an important argument (so I'm not posting it as an answer), but maybe still good to be aware of: in some programming languages, e.g. in C++, the notation $--x$ has a very different meaning than $-(-x)$. Mar 28 at 8:57
• These two $-$ signs have different meanings - one is a binary operation (as in $a-b$) and the other is "unary" (as in $-a$). Thus, I wouldn't be so suprised that there's a convention against putting them one next to the other. Mar 28 at 10:17
• There are plenty of cases where students mess up because of the lack of brackets, and now you want to tell them omitting brackets is ok? No! No! No! check this answer for another case of missing brackets flunking a calculation: math.stackexchange.com/questions/4655389/… Mar 28 at 12:48
• 2-(3-5) has a very different meaning to 2-3-9. So keep the parentheses. it's short for 2-(0-3).
– stan
Mar 28 at 17:38
• I didn't know $17 - -59$ was not acceptable in math. Maybe I forgot. It is acceptable in C and interpreted as subtracting a negative number. It's the same in all the half dozen or so languages I have used. One should take care to differentiate two hyphens without a space and two hyphens with a space (when typing a computer program), because they do mean something different. As for board work, using parentheses makes the expression easier to read. Some people attempt clarity by writing the subtraction and unary minus at different levels thus: $17- {}^-59$ or even with addition $17+ {}^-59$. Mar 28 at 19:21

1. It is not a good idea to against a convention, unless you have strong reasons for it.
2. That said, you might not want to punish those going against the convention, either.
3. An obvious reason for having something between the minus symbols is that when writing by hand (and maybe in some fonts) it might be far too easy to write the minus symbols together, and far too hard to figure out how many minus symbols there are supposed to be in any given case.
4. Also, you would typically want to have the parentheses in cases like $$2 \cdot (-3)$$ for similar issues of legibility that $$2 \cdot -3$$ has. Hard to know whether the dot is supposed to be there or some artifact of the writing process.

It might be a bit of redundancy to have the parentheses, but some redundancy adds legibility. Error correction for human language.

• (3) and (4) seem to be more or less the same (just in opposite directions). It's for sure a valid point! I just wonder if they warrant the problems students just have adhering to the notation... Mar 28 at 13:44
• To add to (3), I must point out that many computer systems will automatically change to consecutive hyphens -- into an en dash – (neither of which is a minus sign). Mar 28 at 19:47
• For sure 3) can easily happen with with pen/pencil. 3-6 can look like 3--6 if a hand jiggles or a pen is low on ink. 3- -6 can look like 3-6 if the-'s are written quickly and happen to overlap slightly ("writing the minus symbols together"). Mar 29 at 13:17

Others have talked about legibility, so I want to address the importance of the parentheses in an expression like $$17\color{red}{-}(\color{blue}{-}59)$$ from the perspective of educational psychology. As @Michał mentions, the two minus signs in this expression have different meanings. To emphasize the difference, we can decode the expression as: $$17\color{red}{\text{ take away}} \color{blue}{\text{ the opposite of }} 59.$$ As @ryang mentions, there are good reasons to pronounce both minus signs as “minus,” and it makes sense that we use the same symbol to represent subtraction and negation, since $$a-b=a+(-b),$$ but this is not a trivial fact. It can be quite confusing for students that a symbol which has always signified subtraction now also signifies negative and negation.1 Using parentheses helps students distinguish the different meanings of the minus sign.

Some curricula attempt to address this by introducing negation using a different symbol, e.g. $$b^* \ \text{or }\ \overline{b}\ \text{ or } {}^−\!b$$ as @Raciquel mentioned. Such a curriculum would then introduce the notation $$-b$$ after students are convinced of the fact that $$a-b=a+b^*\!.$$

Students need to be able to recognize and go between the different meanings of the minus sign to understand negative number arithmetic. In a study by Joëlle Vlassis, she found that when students were asked to solve the equation $$4-x=5$$, some erroneously wrote \begin{align*} 4-x&=5 \\ x&=5-4 \end{align*} which is a very common mistake. In her interpretation, this was because

Students considering, in this context, only the subtraction function of the minus sign could not imagine writing this sign before x at the second stage because, from their point of view, there was no more operation.

Some students who found $$x=-1$$ were unable to write $$4-(-1)=5$$ as justification.

It did not occur to students to put parentheses around the solution... For students who thought that these minus signs were two binary signs, it was inconceivable to write an expression with a succession of two "subtracting" signs.

Difficulty interpreting minus signs can happen when other operations are involved as well. For example, with the equation $$-6x=24$$,

When the interviewer suggested the solution $$-4$$ to them, these students felt that this was not possible, since you would have times minus 4." These two students appeared perplexed about the possibility of producing an expression in which the "times" and "minus" signs followed one another.

So the convention of using parentheses to separate the unary minus from other operations is a thinking tool as much as it is a communication tool. It is natural for students to find it difficult at first, because we are not simply asking them to adopt a new writing convention, but to understand and attend to the different meanings of the minus sign.

1: “Negative” and “negation” are distinct ideas. For example, when $$x$$ is negative, $$-x$$, the opposite of $$x$$, is positive. So for students, there are three ideas related to the minus sign that need to be distinguished: subtraction, negative numbers, and negation. This article (public access through the NSF) goes in depth on the different meanings and has suggestions for teachers on how to help students make sense of the minus sign. It really is difficult!

• The problem with $4-x=5$ is that it's generally solved by moving terms across the $=$ symbol which changes their sign: $4-5=x; -1=x; x=-1$. I suppose that's a misunderstood shortcut for adding $x$ to both sides, etc. Mar 29 at 9:17
• @AndrewLeach Vlassis mentions that, for this equation, the more impactful error was writing $x=5-4$, dropping the negative sign on the left because now there is “no operation” happening. I think this is alleviated if students can flexibly switch between interpreting the minus sign as subtraction and negation, $4-x=4+(-x)$. For other equations, like $-6x=24$, being unable to write $-6(-4)=24$ was more of an issue. Some students claimed that this was impossible, because “you would have times minus four,” indicating that they were stuck viewing minus as a binary operation. Mar 29 at 10:41
• $z^*$, as in the superscript asterisk, might not be such a good choice as it is used to denote a complex conjugate. As is an overbar. Mar 29 at 17:10
• @AndrewMorton I doubt that this notation will survive in class up to the point of introduction of complex numbers. It would be replaced by the usual a* = -a fairly soon, as Justin writes in the answer as well. Mar 30 at 11:36
• TLDR (and I appreciate that we disagree): it is logical and helpful to read negation (unary) and the analogous subtraction (binary) as "minus", without calling on the adjective "negative" (my above link elaborates with the fact that many European languages read -7 as "minus seven" rather than "negative seven", and that you anyway read ±5 as "plus minus 5"). To wit: for $x<−3,$ I would assert, "minus $x$ (e.g., minus minus $7$) is nonnegative," instead of, "negative $x$ is nonnegative." Using the same word in non-identical similar contexts is not unusual in Mathematics (or any language). Apr 2 at 5:39

Are there any reasons why this notation is necessary?

Yes, of course - it is necessary to avoid easily avoidable errors introduced by unclear writing. While it may usually be obvious from context if you see that notation without parantheses in typewritten text (i.e., computers...), this is not the case for pupils writing by hand.

Looking at my own children, writing cleanly was a major obstacle at all times. If they then also are maybe not finding the actual content of their maths trivial, it is just an unneccessary additional cause of error and frustration.

This continues into adult life. I.e. when writing computer code, where a compiler does interpret expressions in a very formal way, with crystal clear semantics, it very often makes sense and is a convention to add completely superfluous parentheses for readability and maintainability.

So aside from teaching the maths itself, it is equally important to teach that it is important to communicate clearly. A correct formula is still wrong if the reader cannot read or interpret it because it's hastily scribbled, and in-built "error correction" techniques like parantheses have also been skipped.

• It is hard to overemphasise the value of clear communication (which also informs clear thinking), even when one doesn't remotely need to write code for a living. The habit of and respect for clear communication is one of the most important things we can inculcate in students. Mar 29 at 15:42

My suggestion:

1. Use the notation yourself
2. Explain why it is helpful
3. Show examples of where it helps to avoid confusion
4. Don't require it from students (i.e., don't mark off for not using it)

If 1-3 are in place, (4) will eventually happen with maturity, even if it isn't this year.

• Disagree on the last point about (4) happening naturally. My experience at a U.S. community college is that many students will blithely make syntax (and code styling) errors in perpetuity if there's no explicit barrier to it. It doesn't matter how many contrary examples they see; not everyone is able to pattern-match automatically. Apr 19 at 17:21