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Okay, so my teacher gave us this to solve: $$a + b/c = 2$$

In this equation he explained that there were many ways to do it. However, we were told to do it with at least two negative numbers. I was a bit stuck at this one, however, more than a couple of students in my class had figured it out anyway.

So one of the solutions to this problem was to do $ -1 -1/-1 = 2$ ( $-$ & $+$ becomes $-$ and then $-$ & $-$ becomes $+$). (Of course assigning $a$, $b$ and $c$ $-1$ as a value). Another solution was to do for example $-5 + 1/-2$ ($a=-5$ $b=1$ & $c=-2$ and the $- / -$ is $+$).

However, there was still something that didn't seem right, although I could see the logic of how these made the answer. But then I realized what was bugging me. The order of operations, which states that you always should do multiplication and division before addition and subtraction (which is something these equations do not follow). So I raised my hand and asked the teacher about this, and he just told me that this wasn't the case for these kind of math problems.

He also told me that he had no logical answer to why it wasn't. However, I also asked another (outside) person about this (whom is exeptionally good at math), and she told me that it didn't matter if it was letters (algebra) or just normal numbers, the order of operations would still have to be used. In which case what I learned in class is wrong. So my question is, what is right and what is wrong? And also why?

If anyone could help med with this it'd be awesome! Kind regards ~

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$a + b / c = 2$ is ambiguous notation.

The order of operations gives us a way to resolve these ambiguities, and the correct thing to do would be to multiply and divide before adding or subtracting $a + (b / c) = 2$ would be the correct way to proceed.

But, better to write it in a way that is clear.

$a+ \frac bc =2$ would be more clear.

As would, $ \frac {a+b}c =2$ although it would have a different meaning. But, perhaps what the teacher intended.

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    $\begingroup$ It is hardly ambigious, it is very clear what it mean. $\endgroup$ – Zelos Malum Oct 18 '16 at 15:36
  • $\begingroup$ No, it is bad, bad, bad. That someone needs to ask the question means that it is ambiguous. And, when you consider how simple it is to make it unambiguous, there is no excuse. $\endgroup$ – Doug M Oct 18 '16 at 15:38
  • $\begingroup$ It is perfectly clear, because others have not done their basic homework doesn't that mean it's ambigious. There are notation that is very ambigious in mathematics and this is not one. $\endgroup$ – Zelos Malum Oct 18 '16 at 15:39
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    $\begingroup$ (+1), but I wouldn't say the notation is ambiguous, since there's a widely-accepted convention that dictates $a + (b/c) = 2$. The notation is, however, almost perversely likely to confuse, i.e., to be (mis-)interpreted as $(a + b)/c = 2$. $\endgroup$ – Andrew D. Hwang Oct 18 '16 at 15:47
  • $\begingroup$ @Doug IF there were parentheses around $a+b$ so that the problem read $$(a+b)/c = 2$$ then it is correct think of it as: $\frac{a+b}{c} = 2$. With respect to $a+b/c =2$, we have one correct answer: $a + \frac bc = 2$The only ambiguity is introduced by humans who mean one thing, but write another thing, or read one way, instead of the way it is correctly written. $\endgroup$ – Namaste Oct 18 '16 at 15:48
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Your teacher is wrong, the order of operation applies always. $$-1-1/(-1)=0$$ as that means $$-1-\frac{1}{-1}=-1-(-1)=-1+1 = 0$$

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  • $\begingroup$ I meantthat, it seems like I missed the word "wrong" for some reason. Thanks for pointing out out! $\endgroup$ – Zelos Malum Oct 18 '16 at 15:41
  • $\begingroup$ Thanks for the correction! $\endgroup$ – Namaste Oct 18 '16 at 15:42
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One should consider $/$ just like the multiplication dot (quite often omitted) as far as order of operations is concerned.

The “in-line” translation of $$ \frac{a+b}{c}\tag{1} $$ is not $a+b/c$, because this would mean $$ a+\frac{b}{c}\tag{2} $$ which is a very different thing. Nobody interprets $a+bc$ as “multply by $c$ the sum of $a$ and $b$”, because the commonly accepted order of operations is that multiplication has the precedence over addition. Similarly, quotient is to be interpreted as “multiply by the reciprocal”, so it has the same precedence as multiplication (and “take the reciprocal” has even higher precedence than multiplication).

If we want to write $(1)$ in-line, we have to resort to parentheses $$ (a+b)/c $$

Thus $$ \frac{2+4}{3}=2 \qquad \frac{-1-1}{-1}=2 \qquad \frac{-5+1}{-2}=2 $$ are correct, whereas $$ 2+4/3=10/3 $$

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"PEDMAS"

Nice acronym for remembering the order of operations, though I'd like for it to be remembered as Parenthes, Exponents, (Division, Multiplication), (Addition, Subtraction), so that there are four levels of precedence:

  1. Parentheses (Innermost to outermost)
  2. Exponents
  3. Division, Multiplication
  4. Addition, Subtraction.

If there's a "tie" in precedence (at level 3. or at level 4.) perform the operations from "left to right."

Your teacher was not correct, though s/he may have been trying to prepare your class for lessons in "the order of operations", by make a point, through the task you describe, that without following them, there is going to be ambiguity and/or confusion in interpretations. The fact that you brought the order of operations up in class may have surprised him/her (i.e., you may be ahead of the game), so s/he perhaps was just trying to put your observation off until s/he could implement the planned lessons. Bad call, on his/her part!.

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  • $\begingroup$ For a worthwhile explanation of the parsing of infix expressions, see [ en.wikipedia.org/wiki/Shunting-yard_algorithm?wprov=sfla1 ]. $\endgroup$ – Senex Ægypti Parvi Nov 18 '16 at 20:23
  • $\begingroup$ @SenexÆgyptiParvi If your want to post an answer yourself, do so. But please stop editing my post. $\endgroup$ – Namaste Nov 18 '16 at 20:26
  • $\begingroup$ @SenexÆgyptiParvi If you have some sort of compulsion to add to my answer, put your contribution in a comment. $\endgroup$ – Namaste Nov 18 '16 at 20:28
  • $\begingroup$ \\ OK, here goes: When consecutive exponentiations are encountered, they are processed from right to left, unlike other types of operations, which are processed from left to right. $\endgroup$ – Senex Ægypti Parvi Nov 18 '16 at 20:34
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You would have to do what is on top of the equation; or you could divide the equation as it is, and still get the same answer. So as you were confused about the order of operations, here is a brief explanation given your equation. Suppose we have your equation, -1-1/-1=2, we could divide at the beginning, so that when -1/-1=1, both of the numbers on the top, we would get the equation 1+1=2 if we were to do your equation through strict pemdas, we would get-1-1/-1=2, so that it becomes 1+1=2. Many people omit strict use of the order of operations because the answer is still equivalent, so if we were to do subtraction first, we would get -2/-2=2 which is a true statement

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