# 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:** 2
**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, 7:28pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/21 "2011-01-24T19:28:05Z")
> [@Indistinguishable](#): > > 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. Dammit. I knew I should have spent more time looking for a counterexample. Ugh. ---
**Author:** ![Angua](https://avatars.discourse-cdn.com/v4/letter/a/8baadc/32.png) [@Angua](https://boards.straightdope.com/u/Angua)\ **Post date:** [January 24, 2011, 10:06pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/22 "2011-01-24T22:06:48Z")
The order that you do the addition in doesn’t matter, provided you remember whether the numbers are negative or positive. Subtracting 8 from a number is the same as adding a number to -8. Thus 16 - 8 +3 = 11 is in fact the same as: -8 +3 + 16 = 11 (-8+3=-5, -5+16=11) Or, what **needscoffee** said ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 25, 2011, 4:27am UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/23 "2011-01-25T04:27:58Z")
Okay, I have another Aunt Sally question. When you’re going around simplifying your equations, let’s say you have a big long thing with something like 3(x+2) in the middle, and of course you go through and make it 3x + 6. Now that no longer has the privileged position of those parentheses, right? ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 25, 2011, 4:36am UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/24 "2011-01-25T04:36:20Z")
To be specific, I really don’t get what I’m doing wrong here and I think it’s some stupid dumbass thing I learned in fourth grade and have happily never needed since. 6(x-2)-3(x+1) =4(3+2) 6x-12-3x+3=20 3x-9=20 x=29/3 Once I got 21/3, I assume I missed a sign. The book says 35/3. Where did I go wrong? ---
**Author:** ![shijinn](https://avatars.discourse-cdn.com/v4/letter/s/cc9497/32.png) [@shijinn](https://boards.straightdope.com/u/shijinn)\ **Post date:** [January 25, 2011, 4:42am UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/25 "2011-01-25T04:42:38Z")
> [@Zsofia](#): > > To be specific, I really don’t get what I’m doing wrong here and I think it’s some stupid dumbass thing I learned in fourth grade and have happily never needed since. > > 6(x-2)-3(x+1) =4(3+2) > 6x-12-3x+3=20 > 3x-9=20 > x=29/3 > > Once I got 21/3, I assume I missed a sign. The book says 35/3. Where did I go wrong? that should be a -3. -3 x 1 = -3 ---
**Author:** ![Tim\_T-Bonham.net](https://avatars.discourse-cdn.com/v4/letter/t/46a35a/32.png) [@Tim\_T-Bonham.net](https://boards.straightdope.com/u/Tim_T-Bonham.net)\ **Post date:** [January 25, 2011, 5:37am UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/26 "2011-01-25T05:37:25Z")
I used to run into this all the time when doing programming walk-thrus. And usually, even a bunch of computer programmers couldn’t agree on this without consulting the manual. My response was to simply say you are NOT ALLOWED to code it that way – you must insert parenthesis so that the order is completely clear, without consulting or remembering anything. Made it much easier to maintain those programs over the years. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [January 25, 2011, 3:52pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/27 "2011-01-25T15:52:08Z")
> [@Zsofia](#): > > I thought I was taught “Please Excuse My Dear Aunt Sally”, which would mean doing all your addition before your subtraction. That’s the mistake right there! The first thing I teach my students if to forget PEMDAS and IMHO, teaching PEMDAS is one of the worst things elementary teachers do because it teaches multiplication then division and addition then subtraction which as you point out is completely wrong. Also, where are roots in that list? I teach my students this chart ( ) ^ r X / - - Order of operations is top to bottom for arithmetic and bottom to top for algebra. If there are two or more operations at the same level in the problem, work them out 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 25, 2011, 4:01pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/28 "2011-01-25T16:01:28Z")
> [@Saint\_Cad](#): > > Order of operations is top to bottom for arithmetic and bottom to top for algebra. If there are two or more operations at the same level in the problem, work them out left to right. What does “bottom to top for algebra” mean? I’m sure it’s something really obvious, but I’m not getting it. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 25, 2011, 4:02pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/29 "2011-01-25T16:02:09Z")
> [@shijinn](#): > > that should be a -3. > > -3 x 1 = -3 Ohhh, thanks. Now that I think about it, I used to make that mistake all the time in school. ---
**Author:** ![mnemosyne](https://avatars.discourse-cdn.com/v4/letter/m/c4cdca/32.png) [@mnemosyne](https://boards.straightdope.com/u/mnemosyne)\ **Post date:** [January 25, 2011, 4:03pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/30 "2011-01-25T16:03:40Z")
> [@Zsofia](#): > > Okay, I have another Aunt Sally question. When you’re going around simplifying your equations, let’s say you have a big long thing with something like 3(x+2) in the middle, and of course you go through and make it 3x + 6. Now that no longer has the privileged position of those parentheses, right? **shijinn** addressed your specific error, but in essence, you’re right - what you are doing when you simplify equations in this manner is putting everything on the “same level” so that values can be easily compared to others, understood, or more simply manipulated to calculate a result (compare how long it would take to plug in x=2 in the original form of 6(x-2)-3(x+1) = y versus in the final form x = (35/3)\*y, for example). In your case you need to remember that the negative sign in front of the three “belongs” to the three, so you need to multiply that sign through as well. The following are all equivalent (unless I made a typo…!): 6(x-2) - 3(x+1) = 6(x-2) +(-3)(x+1) = 6(x-2) + (-1)(3)(x+1) = 6x-12 + (-1)(3x+3) = 6x-12 + (-3x-3) = 6x-12 -3x -3 I hope that helps; sometimes seeing things broken down clarifies problems that can’t be put into words easily. ---
**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 25, 2011, 4:11pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/31 "2011-01-25T16:11:07Z")
> [@Zsofia](#): > > Ohhh, thanks. Now that I think about it, I used to make that mistake all the time in school. Sometimes, if you’re not sure, it helps substituting in a real value to see if you’ve expanded correctly or not. So: -3(x+3) = -3x + 9 **OR** -3x - 9 Well, let’s substitute in and see. Let’s pick 1 as our number Substituting in original equation: -3(1+3) = -3(4) = -12 Our first attempt: -3(1)+9 = -3 + 9 = 6 Our second attempt: -3(1) - 9 = -12 If all else fails, substituting in actual values can help show you where exactly you went wrong. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 25, 2011, 6:43pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/32 "2011-01-25T18:43:31Z")
And if something is written as 4x-(2x-3)=0 (This is in the set of problems at the end of the chapter that sometimes seem like they have stuff not yet covered in them.) I don’t remember how to deal with that, because how can you deal with the parenthesis if you don’t know what x is? ---
**Author:** ![Pai325](https://avatars.discourse-cdn.com/v4/letter/p/c68b51/32.png) [@Pai325](https://boards.straightdope.com/u/Pai325)\ **Post date:** [January 25, 2011, 6:49pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/33 "2011-01-25T18:49:51Z")
You just want to get rid of the parentheses and combine like terms. You will have 4x-2x + 3 = 0 That is the same as 2x + 3 = 0 Isolate the variable by adding -3 to each side, then you have 2x = -3 change the coefficient to one by multiplying each side by 1/2 And now you have x = -3/2 Assuming I read it correctly on my small screen. I can’t tell if it’s -3 or +3 in the original equation ---
**Author:** ![glowacks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/glowacks/32/5548_2.png) [@glowacks](https://boards.straightdope.com/u/glowacks)\ **Post date:** [January 25, 2011, 6:52pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/34 "2011-01-25T18:52:42Z")
Despite my aptitude in mathematics, I had problems with subtraction in the same sort of way; I had assumed that when you subtracted you were supposed to subtract _everything_ after the minus sign. I placed into the “9th grade” algebra class in 6th grade (I actually took a second placement test after apparently acing the first one they gave me; two of my 6th grades classmates complained they should have been placed into my class as well, but were not given the second test; that they had gone to the elementary school run by the same organization whereas I had not was probably a factor) and I was making this mistake because I was not taught how things actually worked. I definitely remember working with parentheses in elementary school, but it apparently had never been a problem before that I was assuming 9-5+4 was not the same as 9-(5+4). When one has never subtracted more than one term outside parentheses before, it is ambiguous until you get told the convention. The convention makes sense, but isn’t necessarily automatic to someone like myself who needs to be told the rules. ## Such a thing never was a problem for division, although the “[obelus](http://en.wikipedia.org/wiki/Division_sign)” (division sign) was generally used, not the “solidus” (slash) that gets used by computers, and it was pretty clear that only the one number after the obelus was supposed to be divided in perhaps partly due to the need to do all divisions first. I personally have issues when seeing things in Excel like A1/A2/B1/B2 - I think it’s quite ambiguous and prefer to write it A1/(A2_B1_B2) - for a similar reason as before: I find the solidus instinctively to mean that everything after it should be by default in parentheses because everything after it is “under the fraction slash” that the solidus sorta represents. Not distributing a negative through the parentheses and other errors involving misplacing signs are extremely common mistakes that lead to computational errors at all levels of mathematics. There’s a definite cognitive bias in forgetting to distribute it; not that I was looking too hard, but I didn’t find the error myself until it was pointed out by someone else. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [January 25, 2011, 6:53pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/35 "2011-01-25T18:53:51Z")
> [@pulykamell](#): > > What does “bottom to top for algebra” mean? I’m sure it’s something really obvious, but I’m not getting it. solve 3X^2 + 2 = 100 Read the chart from bottom to top and you would perform the opposite operation in the order - / r ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 25, 2011, 6:55pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/36 "2011-01-25T18:55:23Z")
> [@Zsofia](#): > > And if something is written as 4x-(2x-3)=0 (This is in the set of problems at the end of the chapter that sometimes seem like they have stuff not yet covered in them.) I don’t remember how to deal with that, because how can you deal with the parenthesis if you don’t know what x is? **Pai325** is correct: you eliminate the parentheses by “distributing the minus sign.” It’s very similar to what happened in your earlier example, where –3(x+1) becomes –3x – 3, except this time the multiplier outside the parentheses is –1 instead of –3. (You can think of 4x – (2x–3) as being equivalent to 4x + –1(2x–3), if it helps.) ---
**Author:** ![Blaster\_Master](https://avatars.discourse-cdn.com/v4/letter/b/cab0a1/32.png) [@Blaster\_Master](https://boards.straightdope.com/u/Blaster_Master)\ **Post date:** [January 25, 2011, 7:00pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/37 "2011-01-25T19:00:22Z")
> [@pulykamell](#): > > What does “bottom to top for algebra” mean? I’m sure it’s something really obvious, but I’m not getting it. Just taking a guess at this one, but I think the reference is to when you’re trying to isolate the variable. If you have 2x + 3 = 0, starting from the bottom of the chart, you’d remove the addition first, then the multiplication second. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [January 25, 2011, 7:15pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/38 "2011-01-25T19:15:07Z")
> [@Thudlow\_Boink](#): > > **Pai325** is correct: you eliminate the parentheses by “distributing the minus sign.” It’s very similar to what happened in your earlier example, where –3(x+1) becomes –3x – 3, except this time the multiplier outside the parentheses is –1 instead of –3. > > (You can think of 4x – (2x–3) as being equivalent to 4x + –1(2x–3), if it helps.) That does help, actually. In general I’m actually finding this shockingly easy, but I keep making those forget-the-minus errors. I remember now being told as a child to imagine a one in there. ---
**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 25, 2011, 7:19pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/39 "2011-01-25T19:19:25Z")
> [@Saint\_Cad](#): > > solve 3X^2 + 2 = 100 > Read the chart from bottom to top and you would perform the opposite operation in the order - / r I see. We’ve never had it presented to us in that way (not that I can remember), but that is the way it works out when isolating a variable. ---
**Author:** ![UncleRojelio](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/unclerojelio/32/3160_2.png) [@UncleRojelio](https://boards.straightdope.com/u/UncleRojelio)\ **Post date:** [January 25, 2011, 7:48pm UTC](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821/40 "2011-01-25T19:48:38Z")
My kids were required to get [graphing calculators](http://www.google.com/products/catalog?q=ti+graphing+calculator&oe=utf-8&rls=org.mozilla:en-US:official&client=firefox-a&um=1&ie=UTF-8&cid=4537577764080176352&ei=dyg_Te3fGYbfgQe4ssXiCA&sa=X&oi=product_catalog_result&ct=result&resnum=1&ved=0CDoQ8wIwAA#) for their first middle school algebra classes. They were taught to enter complicated expressions into them in order to check their work. Later, when they took physics in high school, being able to rapidly key stuff into the calculator was a godsend. You might want to look into getting one. [Previous page](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821.md?page=1) [Next page](https://boards.straightdope.com/t/really-stupid-order-of-operations-question/568821.md?page=3)