# Really stupid order of operations question **URL:** \ **Category:** Factual Questions\ **Created:** [January 24, 2011, 6:49pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821 "2011-01-24T18:49:07Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 24, 2011, 6:49pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/1 "2011-01-24T18:49:07Z")
So I’m going through this “algebra for incredibly stupid people” book so I can help my nephew and niece out with their math problems, and I’m reviewing all that stuff I haven’t touched in literally decades and I’m having a bit of trouble with order of operations. I thought I was taught “Please Excuse My Dear Aunt Sally”, which would mean doing all your addition before your subtraction. I’m getting a few of these sample questions wrong by doing that, though - it seems I’d get their answer if I just went left to right adding or subtracting as I come to it. The book is a little fuzzy on this. I assume it’s right and I’ve been doing it wrong my whole life? For example, for 16 - 2 \* 4 + 3 I get 5 and they get 11. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 24, 2011, 6:54pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/2 "2011-01-24T18:54:01Z")
> [@Zsofia](#): > > For example, for 16 - 2 \* 4 + 3 I get 5 and they get 11. Should be: 16 - 8 +3 = 11 You do the multiplication first, then the rest. ---
**Author:** ![Leaffan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leaffan/32/299_2.png) [@Leaffan](https://boards.straightdope.com/u/Leaffan)\ **Post date:** [January 24, 2011, 6:54pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/3 "2011-01-24T18:54:06Z")
Addition and subtraction are performed in the order that they appear. Always have been as far back as I can remember. Of course multiplication and division come first. ---
**Author:** ![EristicKallistic](https://avatars.discourse-cdn.com/v4/letter/e/85e7bf/32.png) [@EristicKallistic](https://boards.straightdope.com/u/EristicKallistic)\ **Post date:** [January 24, 2011, 6:54pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/4 "2011-01-24T18:54:10Z")
It shouldn’t matter what order you do the adding and subtracting in. It DOES matter when you do the multiplying. So in the example you give, you have to do the multiplication first: 2 \* 4 = 8 Then it’s 16 - 8 + 3 You’re meant to do all adding and subtracting from left-to-right, but as it’s commutative it doesn’t matter. 16 - 11 or 8 + 3 will both get you 11. ---
**Author:** ![Rysto](https://avatars.discourse-cdn.com/v4/letter/r/ecccb3/32.png) [@Rysto](https://boards.straightdope.com/u/Rysto)\ **Post date:** [January 24, 2011, 6:58pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/5 "2011-01-24T18:58:57Z")
> [@EristicKallistic](#): > > You’re meant to do all adding and subtracting from left-to-right, but as it’s commutative it doesn’t matter. 16 - 11 or 8 + 3 will both get you 11. Uh… ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 24, 2011, 6:59pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/6 "2011-01-24T18:59:02Z")
So, how did I get 5 if it shouldn’t matter? (Recall that doing the addition first means you aren’t moving left to right.) ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 24, 2011, 6:59pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/7 "2011-01-24T18:59:21Z")
> [@Zsofia](#): > > I thought I was taught “Please Excuse My Dear Aunt Sally”, which would mean doing all your addition before your subtraction. To clarify: Parentheses Exponents Multiplication Division Addition Subtraction So, go down the list. You have no parentheses. You have no exponents. You DO have multiplication, so you do that next, and get the 16 - 8 + 3 referenced above. You don’t have division. You have addition and subtraction, but their order doesn’t matter, just go left to right. The order of multiplication or division doesn’t matter either, you just have to do it before the addition or subtraction (unless the addition or subtraction is in brackets, in which case they jump to the top of the order.) ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 24, 2011, 7:00pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/8 "2011-01-24T19:00:27Z")
Oh, I see what you’re asking. Hmm…it’s always been left to right for me. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 24, 2011, 7:01pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/9 "2011-01-24T19:01:31Z")
Yeah, I get that. No problem with the rest of it. And now I get that you do the addition and the subtraction together. What I don’t get is the people saying it shouldn’t matter. ---
**Author:** ![Leaffan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leaffan/32/299_2.png) [@Leaffan](https://boards.straightdope.com/u/Leaffan)\ **Post date:** [January 24, 2011, 7:02pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/10 "2011-01-24T19:02:17Z")
> [@Zsofia](#): > > Yeah, I get that. No problem with the rest of it. And now I get that you do the addition and the subtraction together. What I don’t get is the people saying it shouldn’t matter. It does matter, and it’s in the order that they appear. ---
**Author:** ![Rysto](https://avatars.discourse-cdn.com/v4/letter/r/ecccb3/32.png) [@Rysto](https://boards.straightdope.com/u/Rysto)\ **Post date:** [January 24, 2011, 7:03pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/11 "2011-01-24T19:03:36Z")
> [@pulykamell](#): > > To clarify: > > Parentheses > Exponents > Multiplication > Division > Addition > Subtraction > > So, go down the list. > > You have no parentheses. You have no exponents. You DO have multiplication, so you do that next, and get the 16 - 8 + 3 referenced above. You don’t have division. You have addition and subtraction, but their order doesn’t matter, just go left to right. The order of multiplication or division doesn’t matter either, you just have to do it before the addition or subtraction (unless the addition or subtraction is in brackets, in which case they jump to the top of the order.) See, this is what got her confused in the first place. The proper order is: Parentheses Exponents Multiplication/Division Addition/Subtraction BTW **EristicKallistic** , the mistake that you made was that (-1) \* 8 +3 != -1 \* (8 + 3) ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 24, 2011, 7:03pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/12 "2011-01-24T19:03:47Z")
Here’s [Wikipedia’s take](http://en.wikipedia.org/wiki/Order_of_operations): > [@](#): > > These mnemonics may be misleading, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order “addition first, subtraction afterward” would also give the wrong answer. > > The correct answer is 9, which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two. > > There is a new mnemonic featured in Danica McKellar’s books Math Doesn’t Suck[2] and Kiss My Math[3] that does address this very issue: “Pandas Eat: Mustard on Dumplings, and Apples with Spice.” The intention being that Mustard and Dumplings is a “dinner course” and that Apples and Spice is a “dessert course.” Then it becomes not a linear string of operations to do one after the other, but rather the “dinner course” operations are considered together and performed left to right, and then addition and subtraction are considered together, again performed again left to right. > In college mathematics, the rules of priority are (usually) taught correctly, and students are taught the commutative law, associative law, and distributive law, which replace the grade school “rules”. The “left to right” rule is not a law of mathematics. > For example, > > is much easier when worked from right to left, using the associative law of multiplication and treating division as multiplication by the reciprocal. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 24, 2011, 7:04pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/13 "2011-01-24T19:04:21Z")
I think studying up on it is making me forget how I know how to do it. 🙂 If I’d gone with my gut and just skimmed through those problems I would have done it the right way. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 24, 2011, 7:06pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/14 "2011-01-24T19:06:32Z")
> [@Rysto](#): > > See, this is what got her confused in the first place. The proper order is: > Parentheses > Exponents > Multiplication/Division > Addition/Subtraction Yes. I never was taught the mnemonic and always remembered that multiplication is the same level of precedence as division, and addition and subtraction are on the same level of precedence, too. And, in that case, you go from left to right. ETA: Argh…except that it doesn’t really matter with multiplication and division, as 10x4/2 is the same whether you do (10x4)/2 or 10x(4/2). Whereas, 10-4+2 is equal to 10 + (-4) + 2 and not 10-(4+2), which is 10-4-2. ---
**Author:** ![EristicKallistic](https://avatars.discourse-cdn.com/v4/letter/e/85e7bf/32.png) [@EristicKallistic](https://boards.straightdope.com/u/EristicKallistic)\ **Post date:** [January 24, 2011, 7:11pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/15 "2011-01-24T19:11:02Z")
Yeah, please ignore me for saying something stupid. My brain stopped working when I made my earlier post 🙂 ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [January 24, 2011, 7:11pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/16 "2011-01-24T19:11:28Z")
> [@Zsofia](#): > > I think studying up on it is making me forget how I know how to do it. 🙂 If I’d gone with my gut and just skimmed through those problems I would have done it the right way. Just remember that addition IS subtraction. 5 - 8 is the same thing as 5 + -8. So do them in the order they appear from left to right. (Same goes for multiplication and division.) ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [January 24, 2011, 7:12pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/17 "2011-01-24T19:12:32Z")
And it should be “Please excuse my dead Aunt Sally.” ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 24, 2011, 7:20pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/18 "2011-01-24T19:20:06Z")
> [@pulykamell](#): > > ETA: Argh…except that it doesn’t really matter with multiplication and division, as 10x4/2 is the same whether you do (10x4)/2 or 10x(4/2). Whereas, 10-4+2 is equal to 10 + (-4) + 2 and not 10-(4+2), which is 10-4-2. It matters with multiplication and division the same way as it matters with addition and subtraction. Compare (10/4)x2 = 5 to 10/(4x2) = 1.25. The subtlety is that it doesn’t matter if you rewrite everything into just multiplication and reciprocation, same as it doesn’t matter if you rewrite everything into just addition and negation. But what gets negated/reciprocated when you rewrite subtraction/division is highly dependent on how you group things; to properly interpret our subtraction/division notation, you have to avoid clumping things together when negating/reciprocating, which [because we stick the - and / signs on the left of the things they negate/reciprocate] amounts to the same thing as “Go from left to right”. ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [January 24, 2011, 7:21pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/19 "2011-01-24T19:21:37Z")
Yes. Any string of numbers (or terms) added or subtracted gives the same result. Ditto, any strng of terms multiplied or divided gives the same result. Commutative means you can shuffle the order. As pointed out above, 16-8+3 does not mean 16 - (8+3), it means (+16) + (-8) + (+3) For Example: Similarly, you have to be clear on order of operation in multiplication/division. 2_3/5_6, unless brackets of some other rule intervens, means: (2)_(3)_(1/5)_(6). Unless someone states the question differently, it does not mean divide by (5_6), it means divide by 5 and mutiply your result by 6. The trouble is that simple in-line formula writing does not convey the same subtlety as a fancy mathematical equation. This is the source of your confusion. OTOH, the people posing the problem deliberately wrote it this way to ensure the lesson gets across. I bet you’ll never forget it this time… Once you undestand this, it is obvious that any string of addition/subtraction(s) can be done in any order, and any string of multiplication/division(s) can be done in any order. You just have to settle all the multiplication/divisions before tackling addition/subtraction. ---
**Author:** ![Gedd](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gedd/32/4938_2.png) [@Gedd](https://boards.straightdope.com/u/Gedd)\ **Post date:** [January 24, 2011, 7:23pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/20 "2011-01-24T19:23:46Z")
> [@needscoffee](#): > > And it should be “Please excuse my dead Aunt Sally.” Yes, there are so many better mnemonics. For example: Panthers Eat Many Dead Animal Sandwiches [Next page](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821.md?page=2)