I got sucked into the contemporary retread of last year's viral math controversy about 8 ÷ 2(2+2).

I'm on team 1:

8 ÷ 2(2+2) =
8 ÷ 2(4) =
8 ÷ 8 =
1

I oppose team 16, which says:

8 ÷ 2(2+2) =
8 ÷ 2(4) =
4(4) =
16

(See last year's NYT piece for a primer if you like.) https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html

At issue is the standard conventional "order of operations", which school children memorise via a variety of acronyms, but which say that multiplication and division take equal priority, with tiebreakers given left to right. https://study.com/academy/lesson/what-is-pemdas-definition-rule-examples.html#:~:text=PEMDAS%20is%20an%20acronym%20for,until%20the%20calculation%20is%20complete.

The official rules school children learn do predict that 8÷2(2+2)=16.

In signing up for team 8÷2(2+2)=1, I'm saying that those rules are wrong. I think the actual conventions employed in mathematics are at least slightly more complex that the ones school children memorize.

PEMDAS, I gather, was invented by teachers in an attempt to systematize and teach the conventions that had developed in the wild.

It did so imperfectly.

One strange implication of the system children learn is that 8÷4n=2n

(Without brackets, we do division first, and 8÷4=2, then multiply n.)

I think this is a reductio on team 16. https://twitter.com/jichikawa/status/1296892557251641344

But not all agree. Here is someone who embraced that result, and pointed out that online calculators do too. (That was news to me, and interesting.) https://twitter.com/SteveMcRae_/status/1296898694667591680

There are, however, even more absurd implications along these lines.

If you think you need to divide first for 8÷4x, then it seems you should say the same for 8÷xy.

In other words, if

8÷4y = (8÷4)*y = 2y

then

8÷xy = (8÷x)*y = 8y/x

Not only is 8÷xy a completely bizarre way to instruct someone to divide 8 by x and then multiply by y, the same online calculator @SteveMcRae_ cited agrees with me that this is not the standard interpretation. https://www.wolframalpha.com/input/?i=8%C3%B7xy

So what this shows is that sometimes multiplication takes priority over division, even if there are no brackets. It follows that PEMDAS is too crude and simple to capture all cases correctly.

When does multiplication take priority? I think it does when the factors are juxtaposed directly, as opposed to connected with a times symbol. We read those expressions as a unit, and insist on evaluating them together.

Juxtaposition multiplication takes high priority.

(The other choice — which might be what Wolfram Alpha is doing — would be to say that it takes priority with variables but not numerals. This seems like a strangely ad hoc thing to say. (Note that it too departs from PEMDAS, but in a weirder way.))

So where does that put us? It means 8÷4n should be 2/n, not 4n.

(This means at least some online calculators are using the wrong conventions!) https://www.wolframalpha.com/input/?i=8%C3%B74n

It also vindicates team 8 ÷ 2(2+2) = 1.

Do the addition first, then the multiplication, because it is juxtaposed. Division is last in this case.

n.b.: 8 ÷ 2 x (2+2) would be a different story!

More generally, I think this shows us something a little bit interesting about how conventions arise. Sometimes they actually create the rules; other times they attempt to capture rules already implicitly in place. Some feel weirdly in-between.

But adhering too strongly to one's systematisation of a norm can be a mistake — sometimes what look like surprising or counterintuitive results actually just show that one just didn't systematise things right in the first case. https://twitter.com/jichikawa/status/1296872123915358209

(It is interesting to think through parallel dialectics in e.g. ethics and epistemology!)

End of thread. (And I'm off to a meeting, so I'll read replies later.) Thanks to @philoso_foster and @SteveMcRae_ for illuminating discussion along the way.

I like this undated piece I found on a retired Berkeley math professor's webpage. It references a version of this question that went viral almost 20 years ago.
https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html

Bergman takes a more circumspect view than I; he thinks the expression ambiguous. But like me, he's unimpressed with the PEMDAS argument, also linking it, as I'd guessed, to educators of children. (Unlike me, he was a math professor who speaks with some topic-specific authority.)

See also https://twitter.com/RadishHarmers/status/1287486004472029185

Hello beautiful!

https://www.purplemath.com/modules/orderops2.htm#:~:text=The%20general%20consensus%20among%20math,together%20before%20processing%20other%20operations.