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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?

Blue
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RMS
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    If division and multiplication take equal precedence, then the first. – Randall Jul 26 '21 at 21:01
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    If I see that I thwap the author for being imprecise. Randall is right in the usual order of operations with $2B$ being short hand notation for $2\cdot B$. However, it may very well be meant to be understood as a single term $(2B)$. – Alan Jul 26 '21 at 21:02
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    @Salihcyilmaz: it does matter: if $A$ and $B$ are real numbers, $(A \div 2)B$ is only equal to $A \div (2B)$ if $A = 0$ or $B = 1$. – Rob Arthan Jul 26 '21 at 21:10
  • @RobArthan I think he's referencing Alan's comment. $2B$ is $2 \cdot B$ no matter what. – Randall Jul 26 '21 at 21:11
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    @Randall: perhaps you are right. I'll leave my comment anyway. – Rob Arthan Jul 26 '21 at 21:13
  • So we would say $A/B(C+D) = AC/B + AD/B $ ? – RMS Jul 26 '21 at 21:14
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    I would strongly advise against omitting the first pair of brackets in $(A/B)(C + D)$. Note that that left-to-right rule for multiplication when the multiplication sign is elided is dubious. If you see a multivariate polynomial like $3XYZ + 2XY$, it is usual to think of $3$ and $2$ as the coefficients of the monomials $XYZ$ and $XY$, i.e., to read it as $3(XYZ) + 2(XY)$. This doesn't make any difference when you plug in numbers, but it is the "right' reading in abstract algebra. – Rob Arthan Jul 26 '21 at 21:20
  • Some of my students do this and it's really annoying for the exact reason as the question alludes to. – Adam Rubinson Jul 26 '21 at 21:28
  • I think this previous answer of mine can give you some insight into why there is some confusion, though it doesn't give you a definitive answer. – JonathanZ Jul 26 '21 at 21:31
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    At the risk of being flippant you shouldn't ever see $A\div 2B$. No author should ever use it because it is is ambiguous and misleading. – fleablood Jul 26 '21 at 23:18
  • " I think he's referencing Alan's comment. 2B is 2⋅B no matter what" No. It isn't. We just gave an example where it is not. If $2B$ is $2\cdot B$ no matter what then $A\div \color{green}{2B}$ is $A \div (2\cdot B)$. This would be akin to claiming $2+B$ is $2+B$ always and then noticing that $-2+B$ does contain the string "$2+B$" within it. In this case the string does not mean $2+B$; It's just a string within a larger context. – fleablood Jul 26 '21 at 23:24
  • To be flippant again my answer is. By all rules it ought to mean $(A\div 2)B = \frac {A\times 2}B$ but I imagine over 98% of all mathematicians seeing it would read it as $A\div (2B) = \frac A{2B}$. The convention of seeing a specific numeric value $2$ before a variable $B$ and interpreting it as the value $(2B)$ is just too common and too strong for us to not interpret it that way despite ALL the rules of order of operations tell us it is not so. – fleablood Jul 26 '21 at 23:29
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    In mathematics you never write $A \div 2B$, so the question never comes up. Agreeing with @fleablood What you might see in a math paper is $A/2B$ meaning $\frac{A}{2B}$; since if you want $(A/2)B$ you would just write $AB/2$. – GEdgar Jul 27 '21 at 02:04
  • ALso, after the 5th grade you should never really see the the division sign, $\div$, ever again. And $\frac A{2B}$ and $\frac A2B$ are utterly unambiguous. – fleablood Jul 27 '21 at 03:16
  • When I can't tell I usually do it both ways so the grader is aware of the ambiguity. – CyclotomicField Jul 28 '21 at 01:16

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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.

Steven Alexis Gregory
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