Who Decided the Order of Operations?

By Deane Barker on May 13, 2007

I check my 12-year-old son’s math homework every night, and lately he’s been neck-deep in the Order of Operations. This is the order in which you do smaller stuff in a larger problem. For instance:

3 + 5 * 7 / (8 – 7)

In this case, you need to do the 8 – 7 first since it’s in parentheses, then the 5 * 7, and the result should be divided by the 1 from the parenthetical operation then add to the three. So it goes like this:

3 + 5 * 7 / 1
3 + 35 / 1
3 + 35

The answer for the above problem is 38.

The mnemonic for this is:

Please (parentheses)
Excuse (exponent operations)
My Dear (multiplication and division)
Aunt Sally (addition and subtraction)

So, here’s my question —

Why is this the order of operations? This has been so designated as the way we’re supposed to do it, but why? Why does addition and subtraction come before multiplication and division? Who made the rule and why? There are two possibilities:

  1. Someone just decided this and it stuck because we all needed a consistent way of doing it.
  2. There is some higher mathematical truth to it, and at some base level this is the only way to do it because of some core logical principle.

So, which is it? Does anyone know? I’d be interested if this is just an arbitrary (but necessary) decision someone made.

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Comments

  1. it’s (2).

    the reason you do exponents (and roots) first is that they may be broken down into multiplication. likewise, multiplication (and division) are based on multiplication.

    so, say you have 4+3*2. turning the multiplication into addition would make it 4+2+2+2 = 10, which is not what you’d get if you did the addition first.

    the grouping functions (which is listed as parentheses, but which also includes extended radicals of square root signs, fractions with expressions in the numerator or denominator, and absolute values) are there to allow you to bypass these conventions.

    it turns out that, because these rules are built on the hierarchical nature of the mathematical operations, that it preserves the properties that exists with binomial operations: addition and multiplication is still commutative, and so on.

    hope this helped – i’m used to teaching this extended out over several days, and with out the big words.

  2. oh, and as long as you don’t move stuff around, it’s okay to divide before you multiply. in your example above:

    3 + 5 * 7 / 1 3 + 35 / 1 3 + 35

    you could also have done it as follows:

    3 + 5 * 7 /1 = 3 + 5 * 7 = 3 + 35

  3. Why not check your favorite site?

    I looked at that, but I didn’t find anything on the history and reasoning of it.

  4. I think it’s a combination of 2 & 3: the interaction of basic mathematical principles, which are the actual meaning of the calculation, with notation rules.

    The mathematical principles are the relationships among the various operations, such as 5^2 = 55 = 5+5+5+5+5 (by definition). Adding a value to it: 2+5^2 = 2+55 = 2+5+5+5+5+5. Obviously for this to be true, the exponent and multiplication must be calculated prior to addition (otherwise we end up with 49=35=27).

    As charon and Noel pointed out, there are other ways to notate operations. But charon’s proposal with no operation order rules will require larger statements for the same calculation (similar to the relation of Hindu-Arabic numerals to Roman numerals). RPN does work well, but it does so by shifting responsibility for operations order from the person (or machine) performing the calculation to the person (or machine) notating the calculation. So if we write your problem in RPN as 3 5 + 7 * 8 7 – / then we have given the order. 3 5 + 7 * 8 7 – / = ((3 5 +) 7 *) (8 7 -) /. In essense, each operation given in RPN creates parentheses around the two previous values.

  5. Unfortunate that the last comment author’s name is close to Lisajouss, but anyway,

    Deane,

    Take a place value number system like ours, and expand a number like 247.

    2100+410+7

    If you do the operations left to right you get 2047 which is not what you meant. And so the need for order begins and expands logically to what cmadler and iii point out. The extra notation of parenthesis can be a delight for a 12-year-old who ‘wants it MY way,’ but they may have become necessary from the intent of language as in, “I’ll raise you five and double it!”–are you just adding ten or going for broke?

    As for rounding…write out the ten hindu-arabic numerals 0 — 9

    Starting at 0 for ‘nothing’ (which took awhile for us to accept), the median (middlest) is 4.5 so the 4 is on the side of the start of the existing decade and the 5 is on the side of the next decade.

    Hope this helps the 12-year-old.

  6. you know i is also called “BEDMAS” (Brackets, Exponents, Division, Multiplication, Adding and Subtracting.)

  7. I’ve been trying to research this question myself… Check out the following math problems

    1. 10-9+23/2(5+1)

    Old way: 10-9+23 1+23 24 _ = = __ = 2 2(5+1) 2×6 12

    PEMDAS (from left to right): 10-9+23/2(5+1) = 10-9+23/2×6 = 10-9+11.5×6 = 10-9+69 = 1+69 = 70

    There is a HUGE difference in answers. The younger generation (20’s and below, from what I’ve noticed) tends to get 70 and the older generation (30’s and older, unless they are a math teacher) tends to get 2. A substitute teacher, who is a retired science teacher, wondered when things have changed. So, following the two logical thinkings above, which answer is correct, since when, and why???

  8. I’m sorry, the old way became jumbled…

    10-9+23 / 2(5+1) = 1 + 23 / 2 x 6 = 24 / 12 = 2 They were taught to do everything on one side of the division line, do everything on the other side of the division line, then divide

  9. I agree – exponents are based on how many times a number is multiplied by itself. Multiplication is based on how many times a number should be added to itself.. and then the addition and subtraction are basic arithmetics. Even though we do math mentally, we are really breaking down these exponents into simpler problems we can handle, at which point you can do addition and subtractions left to right.

    so 3^2 expands to 3*3 which expands to 3+3+3.

    But if you do problems only left to right, and given an expression 5+3*3, you would do 5+3, but you have disassociated the 3 from the original multiplication (triple addition) problem.

    So I guess it’s a little hierarchy plus convention. Sure, why does it have to be PEMDAS? It could be something else, but as long as we all follow it and get another result, nobody gets upset)..

    Operations are the tools in Math. PEMDAS dictate the grammar of math. :-) Without grammar, any language, being linguistics or programming, would fall apart because there’s no apparent convention around which to do things.

    Dr.Math

  10. things haven’t changed. you are looking at this as two totally different problems. the way it is presented here the answer is 70. but if you expand the fraction bar it changes the entire problem

  11. Yes; I’d say it is more a combination of your options. Firstly, it IS a higher mathematical law that you need to use the operations in the order specified – that is, exponents must come before multiplication of division, etc; but in turn, the latter must operate before + and – in order to get the right answer. But in turn, THIS is so because of the way we read and tend to represent the English language. So there was also some meaning in a consistency of use as well – for all to understand. It is more because of the way mathematicians have chosen to represent it

  12. Well, it seems that this is not use everywhere (unless you are in IT), where I come from only brackets are used to give order…

  13. 4×4+4×4+4-4×4= ill put this into a word problem corruect me if i am wrong

    have 4 dollars put in bank to gain interest of 400% have 16 dollars put in 4 more for total of 20 and wait for 400% increase have 80 have 4 and pay black market friend to quadruple you bank account. have 320

    hear me out. math is a natural thing and really follows one order of operation witch is time. so why do we not read a math problem from left to right the way we do with words. if we conveyed thoughts by grouping certain words together the we deemed as most important…. sentence read like this because dumb

  14. I’m a student and we’re working BIDMAS at the moment. Our homework for Easter was to research why BIDMAS is in the order it is. i quite literally haven’t found anything, but some of the answers above were very helpful. Thank you.

  15. An equation is just a way of expressing a predetermined answer. If we didn’t use order of operations we would just engineer the equation differently to produce the correct answer. So, yes it is arbitrary and simply a way of getting everyone on the same page. BTW I was never taught this so if it is going to be the accepted and universal method then it need to be stressed more.

  16. As for who came up with OoO it was the same dummy who decided which rules of phonics could be ignored or broken and when.

  17. It seems arbitrary to have multiplication first. I would think that, without parentheses or other groupings, a math problem would logically be solved left to right. With parentheses, it becomes different. 4+3×2 would seem to be 14, but with OoO it’s 10. One would think that was why parentheses were employed, to indicate which goes first. If no par, then simple order would prevail. I realize this won’t change, but it has always puzzled me. (I was a good math student).

  18. “old way” vs. PEMDAS = different answers. Where does that leave the “science” of mathematics?

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