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 ~

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 $$2+4/3=10/3$$

$a + b / c = 2$ 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.

• It is hardly ambigious, it is very clear what it mean. Commented Oct 18, 2016 at 15:36
• 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. Commented Oct 18, 2016 at 15:38
• 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. Commented Oct 18, 2016 at 15:39
• (+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 $(a + b)/c = 2$. Commented Oct 18, 2016 at 15:47
• @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. Commented Oct 18, 2016 at 15:48

Your teacher is wrong, the order of operation applies always. $$-1-1/(-1)=0$$ as that means $$-1-\frac{1}{-1}=-1-(-1)=-1+1 = 0$$

• I meantthat, it seems like I missed the word "wrong" for some reason. Thanks for pointing out out! Commented Oct 18, 2016 at 15:41
• Thanks for the correction! Commented Oct 18, 2016 at 15:42

"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