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I always thought division and multiplication have the same order of operation, so it does not matter which of these you do first. But in helping a young student with her math I realized that it does matter.

Example 1: 8 x 5 / 10. 
Multiplication first: = 40 / 10 = 4
Division first: = 8 x 1/2 = 4

Example 2: 6 / 2 * 4
Multiplication first: = 6 / 8 = 3/4
Division first: = 3 * 4 = 12

Trying to find an answer to this I found 2 mnemonics: BODMAS (which seems to say division first) enter image description here

and PEMDAS (which seems to say multiplication first) enter image description here

I would appreciate some clarification here.

Mauritz Hansen
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    They say the same thing. They say that you should do the divisions and the multiplications in that step, in whatever order they occur as you go left to right. – Arturo Magidin Oct 20 '22 at 18:46
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    These are the same conventions, with different words. They are commonly taught in school. M and D should go left to right. In real mathematics, people *should) use parentheses when there is any doubt., – Ethan Bolker Oct 20 '22 at 18:46
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    I would strongly advise avoiding ambiguous notation. It is easy enough to write, e.g., $6/(2\times 4)$ or $(6/2)\times4$ and thereby remove the ambiguity. As to conventions, note that both Excel and Wolfram Alpha parse $6/2\times 4$ as $12$. – lulu Oct 20 '22 at 18:47
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    OK, so to confirm: 6 / 2 x 4 = 12? – Mauritz Hansen Oct 20 '22 at 18:48
  • Thanks for the responses. As for the ambiguous notation: fully agree (being a programmer myself). These are primary school questions intended to be ambiguous so as to specifically focus on the rules for order of operation. – Mauritz Hansen Oct 20 '22 at 18:50
  • Note that the mnemonics don't separate M from D, or A from S. There's a reason for that. They say: first explicit groupings; then exponentiations; then product-and-divisions-at-the-same-time; then sums-and-differences-at-the-same-time. Otherwise, PEDMAS would have separate lines for P, E, D, M, A, and S. – Arturo Magidin Oct 20 '22 at 18:52
  • @ArturoMagidin Sure, and this is how I always understood it to be. It is the (left to right) part that I apparently never learnt. Thing is, (left to right) is also shown next to Addition and Subtraction whereas there it makes absolutely no difference. If I am wrong here then please tell me. – Mauritz Hansen Oct 20 '22 at 18:57
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    Of course it makes a difference. $2-3-2$ if done left to right is $-3$: $2$ minus $3$ is $-1$, and $-1$ minus $2$ is $-3$. But if you do $3-2$ first, then you get $1$. Neither subtraction nor division are associative. And of course $a-(b+c)= (a-b)-c\neq (a-b)+c$. Exactly the same problem as with mutliplication and division. – Arturo Magidin Oct 20 '22 at 19:00
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    $6-2+4$ is $(6-2)+4=8$ but you could read it as $6-(2+4)=0$. Exactly the same ambiguity that you eluded to in your post (though we are more used to interpreting this as $6+(-2)+4$ so it tends not to generate controversy). – lulu Oct 20 '22 at 19:00
  • @lulu exactly: I interpreted 2 - 3 - 2 as 2 + (-3) + (-2), in which case order obviously does not matter as it is all addition. You say 'we are more used to interpreting this as'. But one of these interpretations must be correct and one must be incorrect, right? – Mauritz Hansen Oct 20 '22 at 19:04
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    But if you do that with addition and subtraction, then you should do the same thing with multiplication and division, and interpret $6/2\times 4$ as $6\times (2^{-1})\times 4$, and then there is no ambiguity regardless of whether you do the first product first or second. If there is no subtraction, just addition of additive inverses, then there is no division, just multiplication of multiplicative inverses. Geese and gander and all that. – Arturo Magidin Oct 20 '22 at 19:06
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    And it's not a matter of "correct" and "incorrect". It's a matter of convention. We agree on how to interpret potentially ambiguous formulas ahead of time, to eliminate ambiguity. We can agree on whatever resolution we want, even choosing between contradictory ones, as long as we are both clear on which convention we are using and we use the same convention. – Arturo Magidin Oct 20 '22 at 19:08
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    Always go left to right. So $6-2+4=(6-2)+4=8$ which, as I said, is what you'd get if you interpreted the expression as $6+(-2)+4$. – lulu Oct 20 '22 at 19:08
  • I guess sometimes one uses rules that you are not actually aware of (such as left to right) and operates under assumptions (like the order of division and multiplication does not matter) that work in most cases (university math, 25 years programmer). And then you help a 12 year old and suddenly realize there is a puzzle piece missing. ;-) – Mauritz Hansen Oct 20 '22 at 19:13

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It definitely matters and the reason is associativity. For both addition and multiplication we have that $(a+b)+c=a+(b+c)$ and $(ab)c=a(bc)$ however for subtraction and division we don't because in general $(a-b)-c \neq a-(b-c)$ and likewise with $(a/b)/c \neq a/(b/c)$.

To get rid of this problem in higher mathematics we get rid of subtraction and division entirely and instead add negatives and multiply reciprocals. So if I write $a-b$ what I really mean is $a + -b$ and if I write $a/b$ I really mean $ab^{-1}$. This allows us to use associativity with abandon and avoids any ambiguity about which order we should do things.

CyclotomicField
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