If you see $A\div2B=C$, do you take $A\div 2B$ to be $(A\div 2)B$, or $A\div (2B)\,$

Question

If you see
$$ A \div 2B = C$$
do you take $A \div 2B = (A\div 2)B$ or $= A\div (2B)$? How do I know which one to choose?

Answer 1

I've seen $\log, \ln$, and $\lg$. Also $\log_{10}$ and $\log_e$. But I still hate it when people use $\log$ and they mean $\ln$. I've seen $\arctan, \operatorname{atan}$, and $\operatorname{atan2}$. Why can't we agree on which of $\Bbb N$ or $\Bbb W$ contains $0$? How many times do we have to explain that $\sqrt 9 = 3?$ What does PEMDAS say about $2^{3^4}$? Or $\cos 2 \cdot \dfrac{\pi}{3}$?

Yes, there are rules. Yet sometimes mathematician break them when it makes it easier for them to communicate. They always, in some way, explain what it means to them and why they are doing it that way. No that's not contradictory. Done correctly, it's practical and efficient.

So where did you get $A\div2B$ from? Was it from a C programmer? How about an APL programmer? If you just made it up, what did you want it to mean? A question like that is just begging for context.

My opinion is that most mathematicians eschew unnecessary parenthesis. PEMDAS says that it means $(A \div B)C$. So, unless you meant $A\div (BC)$ don't bother. Also, $``\div"$ is evil and should be avoided. It's main advantage was for the time when type was physically set into presses and $A\div B$ fit on one line while $\dfrac AB$ did not.

Note that $\dfrac A2B$ or $\dfrac{A}{2B}$ or $A \cdot \dfrac 12 \cdot B$ do not leave any confusion about what they mean.