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Had a debate on whether you could do addition/subtraction in any order you want. Specifically, for the following:

$9 - 4 + 3$

We both agree that the answer is 8.

I argue that, by giving addition a higher priority than subtraction (rather than the same priority and going left-to-right), you would end up with $9 - 4 + 3 = 9 - 7 = 2$, which is an incorrect answer, and therefore it matters that addition and subtraction have the same priority.

The other person argues that the order of operations doesn't matter and that it can be done in any order, as by giving addition a higher priority than subtraction, you would end up with $9 - 4 + 3 = 9 + (-4 + 3) = 9 + - 1 = 8$, which is the correct answer, and therefore it wouldn't matter if addition/subtraction had different priorities rather than the same.

I'm arguing that bringing the $- 4$ inside the bracket and then performing it before the $+ 3$ wouldn't be done if addition had a higher priority. The fairly long, debate can be seen here if you want to read it, so I don't paraphrase it all and bias my side too much.

My overall question is who is correct; does the order of operations matter with just addition and subtraction? I'm willing to accept answers for either side, so long as they give a reason.

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Let's clarify:

$$9 - 4 + 3 \color{red}{\ne} 9 - (4 + 3) \tag{1}$$

It appears that you are confusing what is means to group together, or associate, the operations.

  • Yes, addition and subtraction are commutative: The operations can be performed in any order.
  • Yes, addition and subtraction are associative: The terms can be grouped in any order before conducting the operations.

BUT, the mistake in the statement $(1)$ above is that the terms haven't been grouped correctly. The correct way to associate the latter two terms is:

$$\begin{align*}9 - 4 + 3 &= 9 + (-4 + 3) \tag{2}\\ &=9-(4-3) \end{align*}$$

In the original statement $(1)$ at the top of this post, what you have done is introduced a second minus sign.

$$\color{red}{9 - (4 + 3) = 9 - 4 - 3} \tag{3}$$

To conclude, there is no ambiguity to what either $(2)$ or $(3)$ means. But they mean completely different things. The parenthesis, used for grouping in this example, must adhere to the multiplicative property of distribution. If we stick parenthesis into a math statement at will, then we run the risk of completely altering the results. To group items properly, we must make sure that our result conveys the same message - the same order of operations.

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  • $\begingroup$ Still a bit confused. If addition is done first, then isn't the $4+3$ to be evaluated before taking it from $9$, regardless of whether brackets are put in there to clarify? I would have thought that going from $9-4+3$ to $9-(4+3)$ makes no difference if addition is being done first. $\endgroup$ – user290079 Nov 14 '15 at 22:23
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    $\begingroup$ The number is $-4$, not $4$. There's no $4$ in the sum. So yes you can do $-4+3=-1$ first. :) $\endgroup$ – Sam OT Nov 14 '15 at 22:31
  • $\begingroup$ My point is that the brackets aren't being used correctly. The first expression reads "take 9, subtract 4 from it and also add 3 to it." The second statement can be read as "take nine, subtract four AND subtract three from it." They don't logically match. $\endgroup$ – suneater Nov 14 '15 at 22:31
  • $\begingroup$ "take 9, subtract 4 from it and also add 3 to it" - If addition was done first, wouldn't it be "Take 9, subtract the result of 4 + 3 from it", which is equal to the second statement? $\endgroup$ – user290079 Nov 14 '15 at 22:33
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    $\begingroup$ The parenthesis distribute that minus sign to all terms inside. $\endgroup$ – suneater Nov 14 '15 at 22:33
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No, they are done at the same time. $9 - (4+3) = 9 - 7 = 2$. Addition/subtraction is a binary operation, between two numbers. Thus when we say $a + b + c$, we mean $a + (b + c)$ or $(a + b) + c$. Note that, for addition, these are the same since + is "associative" (that's the definition of associativity). So we just write $a + b + c$ as it's unambiguous. Note that $a - b$ is, in essence, shorthand for $a + (-b)$. The same holds for multiplication, but not for division.

$$ (8/4)/2 = 2 / 2 = 1; \ 8/(4/2) = 8/2 = 4. $$

That's why you have to specify the order for division. If you consider subtraction as an operation itself -- not the inverse of addition -- then, in a similar way to with division, it is not associative.

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  • $\begingroup$ Bit confused. You've said that "No", an order of operations doesn't matter, but then said " $9−(4+3)=9−7=2$", which shows that you get the wrong answer (2) by changing the order of operations? $\endgroup$ – user290079 Nov 14 '15 at 22:00
  • $\begingroup$ Ok, yeah, maybe not perfectly clear. By 'no order', I mean it is irrelevant what order you do it in -- ie, the operation is associative. But it's a binary operation on the whole thing. Remember that you really only add negative numbers, not subtract positive ones: $a - b \equiv a + (-b)$. So $$9-4+3 \equiv 9+(-4)+3 = (9-4)+3 = 9+(-4+3)=8.$$ $\endgroup$ – Sam OT Nov 14 '15 at 22:04
  • $\begingroup$ "$9−4+3 ≡ 9+(−4)+3$" Is this true regardless of order of operations? This is mainly the part that I'm looking for a reasoning/explanation on. From 9 − 4 + 3, what would performing all of the addition operations first look like? $\endgroup$ – user290079 Nov 14 '15 at 22:12
  • $\begingroup$ It's literally just that saying $-x$ means $+(-x)$. Then you can see it's just a string of three $+$s, and it doesn't matter which order. $\endgroup$ – Sam OT Nov 14 '15 at 22:13
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    $\begingroup$ @smiley You wrote $2/2=2$ $\endgroup$ – cheesyfluff Nov 14 '15 at 22:29
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I would say that the order (for adding/subtracting) doesn't matter AND you need not elevate 'priorities' so long as you realize that subtracting is just adding negative numbers...

So the 9 - 4 + 3 isn't 9 - (4 + 3) but rather is 9 + ( -4 + 3 ).

Putting the negative sign in front of the parenthesis implies that you want to make everything inside negative and that's not what we actually wanted.

So just make everything a signed number, and do all processes as an addition in whatever order is easiest for you.

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It does matter because it reflects the convention of which procedures are first performed to evaluate a given mathematical expression.

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