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Lately, equations such as X=8/2(2+2) have begun being presented on social networks as "very hard problems that will divide the audience just like the striped dress optical illusion".

Of course, most of people coming at different results are blindly following the BODMAS or PEMDAS mnemonic incorrectly, saying that multiplication always goes before division or the opposite, where in reality most sources I've found say "multiplication and division" and "addition and subtraction" are single steps.

I thought those were easy to disprove, but then someone stated that the "left to right" rule is only spelled out for sums and subtractions, and not for multiplication and division, and people writing expressions always needs to use parentheses or horizontal fracional bars to disambiguate an otherwise unknowable order, making 8/2*2=?

Someone else said that implicit multiplications like 2(2+2) must be done first, and that 8/2*(2+2)=4*4=16 but 8/2(2+2)=8/2(4)=8/8=1.

Is there a true consensus on what the correct order is?

Jyrki Lahtonen
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Zachiel
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    Left to right is generally agreed upon among legitimate sources: if you consult wolfram alpha or your phone calculator, that is the system used. However, this should never have to be used. If you are writing an blatantly simple equation that is ambiguous to readers, you have made a mistake. – Kraigolas Aug 06 '20 at 18:52
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    I warmly recommend and fully endorse Gerry Myerson's answer to the linked question. – Jyrki Lahtonen Aug 06 '20 at 18:54
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    I ought to go ahead and bookmark that question since duplicates of it appear so frequently – JMoravitz Aug 06 '20 at 18:55
  • I think this equation lacks a bit. one can assume that the denominator is 2 or [2(2+2)].. Brackets make a huge difference – Omar Shaaban Aug 06 '20 at 18:56
  • I consider that this notation is not so mathematical. I adhere to the fraction bar representation. I happen to use the slash in exponents, and always make sure to parenthesize to avoid any ambiguity. Without that, left-to-right order. –  Aug 06 '20 at 18:59
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    As an aside, I suggest following the rabbit hole and looking at this further related question. The most important takeaway from this for you should be that the notation is ambiguous and if you insist on using it in a particular setting, e.g. in a block of code, you should be triple sure that you know the way in which it will evaluate in that programming language. In maths I would never be caught dead writing p AND q OR r but I see p && q || r all the time at work. – JMoravitz Aug 06 '20 at 19:01
  • I second @JyrkiLahtonen on his opinion. Gerry provided a comprehensive and elegant answer – Omar Shaaban Aug 06 '20 at 19:01

2 Answers2

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If someone ever writes "evaluate $x= 8/2(2+2)$" or something of that nature, my response is: "the question you have asked is ambiguous. Did you mean: $$ x = \frac{8}{2(2+2)} $$ or, $$ x = \frac{8}{2}(2+2)." $$ It's not that math is unclear, it's the person asking who has made an unclear problem (and then going on to claim it's a hard problem). It's like if I asked "does the equation $x^2 + 1 = 0$ have any solutions?" The question is unclear because I should've specified if I am considering only real or complex solutions.

ccroth
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There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible for anyone to mistake the meaning. If they mean, $(8÷2)*(2+2)$, they'll write it that way; if they mean $8÷(2*(2+2))$, they'll write it that way.

Quoted from What is 48÷2(9+3)?, this question is ambiguous. As said, there are two ways to interpret this: $(8÷2)*(2+2)$ and $8÷(2*(2+2))$. If you have any expression $a÷c*b$, it is ambiguous. Implicit multiplication makes it even worse, not specifying the multiplication makes it more ambiguous than it already is.

  • Well, I think it's a pity that there's no Supreme Court for mathematical notation, it would really help. Not even an ISO standard? I also find it weird that nobody found this problematic, but... well, we will need to live with it I guess. – Zachiel Aug 07 '20 at 12:26