Order of Operations Explained: Addition, Subtraction and Conclusions

This is the 6th and last in the series The Order of Operations Explained.
For the other articles in this series, click here to visit the introduction.

I started this series over a month ago. In that time, I’ve gotten pretty deep in thinking, learning and reading about the order of operations. I’ve seen a variety of ways people view, use and teach it.

Before I go too far into some conclusions, though, let’s look at addition and subtraction.

Subtraction is the same as addition.

Yup. You might remember that from the fourth article.

Consider the problem . Moving from left to right, and doing both subtraction and addition as we come to them, we get 4. If we found a book, or person, that meant the full-on PEMDAS and wanted addition done strictly before subtraction, then we would end up with 0. The latter is because we would do the addition of 3 and 2 before we did the subtraction.

Which is right?

It depends on what you really mean. If you don’t know if you should go left to right or strictly addition before subtraction, either look in the textbook you’re using or demand parenthesis.

The text will clearly outline the order of operations it’s following. Be careful, too because there isn’t always agreement among textbooks. I have seen some texts that instruct the learner to do multiplication first and then go back and do all the division signs. While others (and this is more common, today) have us do multiplication and division from left to right, simultaneously.

If you compare contemporary texts to each other, you’re likely to find them all the same. But grab a math text from the 80s at Half Priced Books. I’ll bet you’ll find at least 50% of the time they put division strictly after multiplication. (I’ll verify this the next time I’m there.)

The order of operations needs context.

I have $5 in my bank account. Then I bought a coffee for $3 and a bagel for $2. I might accidentally write down . I still mean, “I need to add up the stuff I spent and subtract it from my balance.” I wrote it in error, though. What’s “mathematically” correct is .

But you knew what I meant.

This was a typo that was helped along by using the context.

Until there’s a reason to do arithmetic, the order in which we do things is arbitrary. If we all agreed to do addition first, then multiplication, we would calculate  and come up with 35 (instead of 23).

As long as we all come up with the same thing, we’re fine.

“We” have agreed to do multiplication things before we do addition things. So “we” would come up with 23 in the example.

Coach G noted it correctly: the order of operations is a convention. In other words, we’ve decided on it. We invented it.

How can you use this to teach your children?

The coolest thing is that you can let them play. Get dirty. Break it.

Remember opposite day? Have that. Let your little one make new rules. Let them see what happens if you all decide one day to do multiplication before addition. If your child is older and doing some algebra, this will mean reversing the order in which you UNDO the operations too!

This is a real brain stretcher. But it’s just math. You’re not building a bridge or balancing your checkbook. Let them break it. Let them see what happens if you make your own rules.

And then they’ll really learn!

Let me know how it goes – share your stories in the comments.

A big thanks to Jimmie’s daughter’s order of operations mnemonic for helping me get it in gear to get this series written.

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16 Responses to Order of Operations Explained: Addition, Subtraction and Conclusions

    • Thanks for writing, Jam.

      There is no difference in the value of those two. However, addition and subtraction are actually binary operations, meaning they work on two things (not three), so technically 3+4-2 isn’t “doable.”

      Of course, we know it is. But only because we’re humans and we don’t need such formalities. At least not all the time.

      Thanks for stopping by!

  1. So… I learned that though addition is before subtraction in “pemdas”, addition can still has to come after subtraction when the problem is: 5-3+2. The only way addition would come before subtraction, is if there were parenthesis around the “3+2” like: 5-(3+2). The correct answer to 5-3+2 is and can only be 4, since there aren’t any parenthesis… Enough thinking 🙁 bye! I will never return, bye!!

    • Thanks for stopping by, Dessy. Sorry to hear you won’t come back – but you won’t ever stop thinking. And yay for that!

    • You error in saying that doing addition first in that statement 5-3+2 = 0 is that the 3 is a negative number. It is the same as saying 5 + -3 + 2 which equals 4. By adding 3 and 2 to get 5, you are making a fundamental error in the problem.

    • Good question, Corbin. It’s just sloppy talk.

      Technically it would be going in reverse order, or undoing what you’ve done. Not really both.

      Thanks for stopping by!

  2. Actually, you made an error. 5-3+2.
    If you did the addition first, you would have -3+2, you can not ignore a number’s sign. -3+2 = -1. 5-1 = 4. Sometimes this is easier for people not to mess up if they write out all subtraction as it truly is (an addition of a negative number). 5-3+2 is shorthand for 5 + (-3) + 2. Easy to forget.

    • Thanks for your comment, Kendra.

      I don’t believe I made an error. My point is that you could (and students often do) look at addition separate from subtraction. If that’s the case, then “doing addition first” would mean that the minus sign doesn’t have the meaning “negative 3.”

      It’s a subtle difference that math teachers, or even people well versed with math, can’t see – we’re too unconsciously competent.

      But math students who haven’t mastered it don’t see the subtraction as “negative addition.”


  3. I disagree with you and Coach G about the order of operations being convention. I’ve noticed this a lot among math educators. They seem to think that the order of operations is just an arbitrary set of rules that people just decided some day, and the world jumped on the bandwagon. THIS IS NOT TRUE! The order of operations is NOT arbitrary. There is a REASON that we do multiplication before addition, and that is because multiplication IS addition: It’s iterated addition.
    Example: 4*5 is really 4+4+4+4+4, which is 20. So given a problem 3+4*5, we MUST do 4*5 first, because the problem must be re-written as 3+4+4+4+4+4 (which yields 23). The problem CANNOT be written as 7+7+7+7+7, which is what you would get if you had done the 3+4 first.
    Summary: the order of operations is NOT arbitrary. It is NOT convention. We as mathematicians didn’t suddenly “agree” on an order. The order of operations is the way that it is because it’s mathematically accurate!

    • I think some would disagree with you on the face that multiplication is iterated addition. (I’m not one of those, though.)

      Regardless, if you want to sit and have a glass of wine someday and play, we could certainly work out a number system where we switch around the order of operations and prove a bunch of interesting things.

      In fact, for our real world purposes, we do need them the way they are. But not all of math is real world. Thus, the convention and arbitrary-ness of it.

    • But you’ve just written it as 7+7+7+7+7!

      Why don’t people just write the things in order? Put the multiplication first, 4*5+3 and it’s perfectly clear. Why mix them up in the first place and then introduce a rule to “un-mix” them?

      By the way, I’m really not having a go at you or saying that you’re wrong in anyway, it’s just that the system seems a bit nuts.

  4. Could you cite a credible source (preferably third party (in lieu to another link on this site)) that would prove that you do addittion before subtraction or you would do multiplication before division?

    • There is no proof that you would do addition BEFORE subtraction. If a book (or set of people) decided that you should do that, then you would need to follow that convention.


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