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I just introduced the concept of adding and subtracting negative numbers to my students and I found that I cannot come up with a good explanation of the use of bracket. In particular, I ask my students to 'add $-3$ to $2$', some of them give me: $$ 2 + - 3$$ I told them to add a bracket like this: $$2 + (-3)$$ but they start asking why. In fact, when I recall my maths lesson in high-school, we seldom, or even never, allow the four signs of operation ($+$, $-$, $\times$ , $\div$) to be 'next to each other'. For example, $8 \times -3$ should be written as $8 \times (-3)$. But I was never told the reason behind this.

Of course, if we are dealing with $2+3 \times 4$ and $(2+3) \times 4$, the necessity of including the bracket is obvious. But there seems to be no ambiguity in the cases I mentioned above. So why do we have to include the bracket? Is it a rule? Or is it just a practice we follow? Thanks in advance.

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  • $\begingroup$ One example of the unary negation operation which might be ambiguous is $-3^2$, which might be intended to mean $(-3)^2=9$ or $-(3^2)=-9$. To avoid this issue, some people developed the habit of always using brackets around negative numbers in expressions $\endgroup$ – Henry Jun 27 '17 at 9:37
  • $\begingroup$ It's just habit and convention. There would be nothing wrong in writing $2+-3$ (and, by the way, $\rm\TeX$ typesets this correctly). Actually, the standard convention is that $2-3$ is a shorthand for $2+(-3)=2+-3$ (whichever you prefer). $\endgroup$ – egreg Jun 27 '17 at 9:40
  • $\begingroup$ Due to the ambiguity of the $-$ (minus) sign: binary operation: $2-3$ vs unary function: $-2$. A binary operator must be enclosed between two "terms": thus, $3+ \times 2$ is not correct. $\endgroup$ – Mauro ALLEGRANZA Jun 27 '17 at 9:46
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    $\begingroup$ For clarity. And (in the case of multiplication) because students who write $2 \cdot -3$ in one step of a computation often accidentally write it as $2-3$ in the next step... (Yes, I've seen this happen many times when grading calculus tests.) $\endgroup$ – Hans Lundmark Jun 27 '17 at 9:58
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The best explanation (beyond "it's tradition") I have is that it's for typographic clarity. If you write $5- -8$ or $5+-8$, there's a risk that you will come back and read it as $5-8$ or $5+8$, seeing the minus as just an extension of the horizontal line in the $+$.

You can get out of that by adding extra space, but it is easier simply to put in a set parentheses.

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I also struggle with the same problem, that is explaining this to students who are just getting aquaintance with the beautiful world of mathematics. I usually try to think about this in an algebraic sense. That is the set $\mathbb{Z}$ together with the usual addition $+$ and $0$ as the unit element constitutes a group. In this sense there is no "minus" as an operation. Every element has to have an inverse and these are simply the "negative counterparts" that is $$ 5-8 $$ is not correct syntax since we only allowed to add elements together since (right now) this is the only operation defined on the set $\mathbb{Z}$. We dont subtract, we "add the inverse of $8$" to $5$ $$ 5+(-8). $$ Using this "more abstract" way may seem way too complex for students in high school but following up this approach one may easily explain other, similarly strange concepts like why is the product of two negatives is positive. To answer to original question the parentheses are there to signify that the minus sign and the $8$ are "bound together". In my opinion we do not like these operations "next to each other" is simply because of readability. I hope this helped :)

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To supplement the existing good answers and comments, here's one reason (admittedly ex post facto rationalization) to justify parentheses:

The unary minus operator is arithmetically equivalent to multiplying by $-1$. Consequently, $2 + -3$ may be viewed as short for $2 + -1 \times 3$. Multiplication binds more tightly than addition.[*] is Out of politeness to the reader, we write $2 + (-1 \times 3)$ to emphasize operator precedence. The parentheses persist even after simplifying: $2 + (-3)$.

[*] This construct is an excellent reason for multiplication preceding addition: If addition came before multiplication, a clever student could write $$ 2 - 3 = 2 + -1 \times 3 \stackrel{!!}{=} (2 + -1) \times 3 = 3. $$

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