Does the order of operations matter with just addition and subtraction?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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