I received this text from my brother: Pam, who created the “Order of Operations” rule? Well, that’s a curious question. Why is he asking? I’d never thought about who “created” these rules. They just kind of made sense to me. Before I did any research, I thought for myself why these rules made sense to me. Here’s my response to my brother.

Suppose I needed to calculate 4 + 3 * 5. Without the order of operations, I would just make each calculation as I come to it. In this case 4 + 3 = 7 and 7 * 5 = 35. But, 4 + 3 * 5 is equivalent to 3 * 5 + 4, right? Because of the commutative property of addition, the order that we add numbers in doesn’t matter – we get the same result either way. So if that’s true that 4 + 3 * 5 = 3 * 5 + 4, then both calculations should give us the same result. But, if we don’t have conventions around this then the second expression, 3 * 5 + 4, would result in 19. Clearly that’s not equivalent to 35.

So why would we choose to multiply first instead of adding? We can think of the expression above (4 + 3 * 5) to mean that I am adding 4 onto 3 bunches of 5. I guess I think that it’s kind of implied that I would want to know what 3 bunches of 5 are before I add 4 onto that number. That’s why we would multiply before we would add.

There is a similar argument to be made with division. Suppose I come across 3 + 1/2. Without our order of operations, we might conclude that this was equivalent to 4/2 (or 2). But isn’t 3 + 1/2 the same as 3 and a half, or 7/2 (which is clearly not the same thing as 2)?

The PEMDAS mnemonic is a little misleading, too. It suggests that multiplication takes precedence over division and that addition takes precedence over subtraction. That’s not true. Multiplication and division are at the same level as are addition and subtraction. For example, if you want to calculate 6 * 3 / 2, you can first calculate 6 / 2 and then multiply that result by 3. Or, you can calculate 6 * 3 and then divide that result by 2. Either way, you end up with a result of 9. You can reason similarly with addition and subtraction.

Personally, I think that the real question is “Who invented parentheses?” I mean, to show grouping by using symbols is just genius. That would change the outcome of the original expression, right? (4 + 3) * 5 is very different from 4 + (3 * 5). At some point, mathematicians agreed that they didn’t need to write the parentheses around 3 * 5 (maybe for the reason that I stated above, maybe not), but that they would need to explicitly group (4 + 3) if that’s what they meant.

My Google search found this response to your question from Ask Dr Math.

Turns out there’s this Facebook post that asks you to calculate 6+1*0+2/2. Some people say the result is 7 and others say it is 1. Which is correct? That’s where the order of operations comes in. And that’s what was behind my brother’s question.

Regarding the parentheses, I have a copy of “A treatise on Algebra” by Colin Maclaurin (of the Maclaurin Series) dated 1756 in which he uses a bar on top of a grouped list of terms, like

______

a+2b+c x d, where we write (a+2b+c) x d, or d(a+2b+c)

so the parentheses came later.

A good reason for parentheses is if you consider 23 x 10 then decide to separate the ten into 6 + 4 it is essential to keep the 6+4 as a single “thing”.

Regarding order of operations it is worse with addition and subtraction.

Not only can confusion arise with 16 – 3 – 5 but also with 16 – 3 + 5

There has to be a rule, left to right or right to left, so long as we ALL agree, but if we see the bits in these expressions as what to do with each number,

and put the + sign in front of the 16, we see that -3 means “subtract 3 from” and +5 means “add 5 to”, in each case to some total, then order for + and -ceases to matter.

I also saw the bar notation in the Ask Dr Math post. I wonder when the standard was changed to parentheses. Do you think it might have had something to do with programming computers?

As for your comment: “There has to be a rule, left to right or right to left, so long as we ALL agree” I don’t agree with this statement, unless I misunderstand what you are saying here.

With 16 – 3 + 5 I can first add 16 + 5 and then subtract 3, or I can compute 16 – 3 and then add 5. Since 16 – 3 + 5 is the same as 16 + (-3) + 5, and addition is commutative, I can add these numbers in any order that I want – whatever order is most convenient for me. And maybe I can even add them more quickly in my head if I can move them around like this and group them. Isn’t that what we want our kids to learn, not that you have to add & subtract in order from left to right or right to left?

I appreciate your reply of why it makes sense. This will help me with my teaching. 🙂

Thanks, Holly. It was fun to think about.