# Confusion about order of operations

The question is simple, why the following is wrong!?

24 $\div$ $\frac{9}{-3}$ = 24 $\div$ 9 $\div$ -3

• It is heavily implied with the way you wrote it that you have $24\div (\frac{9}{-3})$. Note that Division is not associative. Unlike addition and multiplication, generally you have $(a\div b)\div c \neq a \div (b\div c)$. As such, writing $24\div 9\div (-3)$ is ambiguous whether you mean $(24\div 9)\div (-3)$ or if you mean $24\div (9\div (-3))$. As it so happens, the first is wrong but the second is correct. – JMoravitz Aug 25 '16 at 2:13
• i agree totally but I didn't but parentheses to $\frac{9}{-3}$ in my question – Klaus82 Aug 25 '16 at 2:15
• the $\div$ symbol is a horrible piece of notation that should be avoided precisely for this reason. It is ambiguous in how it should be interpreted. – Doug M Aug 25 '16 at 2:21
• If $a-b-c$ is not ambiguous, then why should $a \div b \div c$ be? Isn't it logical to work from left to right? If $a-b-c=(a-b)-c$, the for the sake of consistency, $a\div b\div c=(a\div b)\div c$. – Batominovski Aug 25 '16 at 2:41
• @Batominovski The only reason $a-b-c$ is unambiguous is that people have agreed to the convention that it means $(a-b)-c$ and not $a-(b-c)$. (Nowadays, we "agree" to this convention only in the sense that it is drummed into us in school and we get bad grades if we disagree.) People have not, as far as I know, agreed to such a convention for division. – Andreas Blass Aug 25 '16 at 3:09

## 4 Answers

$24\div\frac{9}{-3}=\frac{24}{\frac{9}{-3}}=24\times\frac{-3}{9}=24\times(-3)\div9$

Division, unlike addition or multiplication, is not associative.

So the result of: $$a \div b \div c$$

will depend on the order you perform the operations in. Thus in most cases

$$a \div (b \div c) \ne (a \div b) \div c$$ In your case, I think you have implied:

$$24 \div (9 \div -3) = (24 \div 9) \div -3$$

which is not the case.

$a \div \frac{b}{c} = a * \frac{c}{b}$

So $24 \div \frac{9}{-3} = 24 * \frac{-3}{9}$

Fractions are thought of as being a single object. Here is something a little bit more complicated

$\frac {x^2 - 3x + 2}{x - 1}$

We evaluate everything above the line together and everything below the line together. No parenthesis are written but they are implied.

So,we see the $\frac 9{-3}$ and reduce the fraction before moving along.

$\div$ happens left to right.

Most math beyond middle school gets rid of the $\div$ notation precisely because it is frequently ambiguous.