# There can be only 1August 1, 2019 10:59 AM   Subscribe

One!

(Can't do a First!)
posted by buzzman at 11:03 AM on August 1 [1 favorite]

Internet divided by question = x
posted by Nancy Lebovitz at 11:08 AM on August 1 [5 favorites]

but is it dividing the internet before or after the evaluation of the brackets?
posted by allegedly at 11:09 AM on August 1 [4 favorites]

Ugh. PEMDAS vs BODMAS is the lamest fight ever. Metric vs imperial is a PHD worthy discussion compared to this.
posted by The_Vegetables at 11:09 AM on August 1 [20 favorites]

The answer is, no one who understands mathematical notation would write that formula, because it is ambiguous and the whole objective of notating a formula in mathematical notation is to avoid ambiguity.

The second answer is, no one really uses the ÷ symbol, except in elementary school. Again, the reason is: It is ambiguous.

If you write 6 ÷ 3 then everyone is clear on what you mean. But as soon as you start stringing together things like 6 ÷ 2 X 4 or whatever, it becomes ambiguous and confusing. So just don't do that.

6/2 X 4 is perfectly clear, as is 6/(2X4). 6 ÷ 2 X 4 is just confusing. I've seen a lot of mathematicians write a lot of things, but never anything like that. Because, confusing.
posted by flug at 11:10 AM on August 1 [129 favorites]

This is so silly. We can all agree that (2+2) is in parentheses, so it binds first. This gives us 8/2*4. Do you believe, like some programming languages, that multiplication implicitly binds before division? Or that the proper order of operations is right-to-left? Then the answer is 1. If you believe, like some OTHER programming languages, that division and multiplication have equal precedence, but that the order of operations is left-to-right, then the answer is 16. But there's nothing special about it, it's just a disagreement about how things are done.
posted by ubiquity at 11:10 AM on August 1 [27 favorites]

posted by mrgoat at 11:11 AM on August 1 [47 favorites]

It's blue and black.
posted by RobotHero at 11:13 AM on August 1 [17 favorites]

I'm so thick I can't work out the question, let alone the answer.
posted by peepofgold at 11:14 AM on August 1 [6 favorites]

42
posted by TedW at 11:14 AM on August 1 [11 favorites]

posted by grumpybear69 at 11:16 AM on August 1 [33 favorites]

Feh. We should be drilling C operator precedence rules into third graders' heads. And maybe covering LaTeX equations in fifth grade.
posted by RobotVoodooPower at 11:17 AM on August 1 [6 favorites]

8 + 2 * (2 + 2)
8 + 2 * 4
8 + 8
16

Disclaimer: I may need glasses
posted by flamewise at 11:18 AM on August 1 [20 favorites]

This is why prefix notation is the One True Way and infix notation is of the devil.
posted by asterix at 11:18 AM on August 1 [16 favorites]

Is anyone else bothered by the fact that they're calling this thing an equation despite the complete lack of an '=' and a second side to which the expression would equate? No? Just me? Alright, carry on
posted by Mister_Sleight_of_Hand at 11:19 AM on August 1 [57 favorites]

This is why postfix notation is the One True Way and infix notation is of the devil.
posted by zengargoyle at 11:19 AM on August 1 [15 favorites]

Do you believe, like some programming languages, that multiplication implicitly binds before division?

I didn't realize that there were some programming languages that handled multiplication before division (instead of at the same priority but going left to right)! Which ones are these? I happen to write a lot of SQL at my job so I'm curious now...
posted by andrewesque at 11:22 AM on August 1 [1 favorite]

I swear - if this turns out to a nation state driven campaign to further divide us...i'll...i'll... be totally not surprised. So *sigh* I guess I'm team sweet 16 for when the purge comes and we need to know who is friend and foe based on mathematical order of operations.
posted by inflatablekiwi at 11:25 AM on August 1 [2 favorites]

Is anyone else bothered by the fact that they're calling this thing an equation

Well, it's consistent with the general rigor-less aura surrounding the entire question.
posted by wildblueyonder at 11:25 AM on August 1 [18 favorites]

Order of operations is the broader term for the PEMDAS vs BODMAS issue
posted by XMLicious at 11:26 AM on August 1 [2 favorites]

There is a space between the "÷" and the rest of the equation; so it has to wait in line.
posted by buzzman at 11:27 AM on August 1 [4 favorites]

Isn't that an expression, not an equation?

[on preview -- yes, Mister_Sleight_of_Hand]
posted by theory at 11:27 AM on August 1 [5 favorites]

Too much time in math spent worrying about ambiguous irrelevancies like this, not enough time spent determining what, exactly is the O'Reilly Factor
posted by COBRA! at 11:28 AM on August 1 [3 favorites]

I don't see a specific rule for this in PEDMAS or BODMAS but there should be a difference in binding between the multiplication-by-juxtaposition notation and explicit ×.

a / bc = a / (b × c)

a / b * c = (a / b) × c

Computer notation doesn't usually support the juxtaposition notation, so it's tempting to rewrite the former as the later. But I would argue it is a mistake.
posted by sjswitzer at 11:33 AM on August 1 [8 favorites]

With apologies to the fine folks who write those ANSI specs, the actual order of operations in C/C++ is as follows:

1. Multiplication and division from left to right.
2. Addition and subtraction from left to right.
3. Put parenthesis around everything else.

I also want to add here that as soon as I saw the OP I knew it must have something to do with the Devil's Operator: ÷.
posted by Horkus at 11:35 AM on August 1 [2 favorites]

I evaluate by PEMDAS (so I get 1), but Android's built-in calculator app gives me 16. Since I had no idea that alternate evaluation procedures are commonly accepted, I had no reason to expect that, and that seems really problematic. Now I'm wondering if calculator apps in different regions give different results?

To be honest I hate things like this. Incompatibilities in measuring systems, evaluation procedures, car lanes, electrical voltage and frequency... they'd be so hard to unify, but nothing good comes of the differences, and sometimes really bad things happen!
posted by trig at 11:35 AM on August 1 [2 favorites]

I’m so happy that I got even one of the correct answers! (1)
posted by sallybrown at 11:35 AM on August 1 [2 favorites]

This is why postfix notation is the One True Way and infix notation is of the devil.

postfix notation One True Way is infix notation of the devil is and
posted by delicious-luncheon at 11:40 AM on August 1 [30 favorites]

Oh yeah, internet? Well check this shit out:

This statement is false

OMFG YOU GUYS IS IT TRUE OR FALSE IDK MIND BLOWN THERE IS NO GOD
posted by echo target at 11:40 AM on August 1 [15 favorites]

This is why I use an RPN calculator.
posted by doomsey at 11:41 AM on August 1 [10 favorites]

I'm struggling to find an analogy for why this whole thing is so stupid... if I ask someone with a poor grasp of German to write a German sentence, is it a German language exercise to understand what they wrote?

I certainly hope not. This is just a weird exercise in how to choose interpretation conventions for a poorly constructed expression. This is not a math problem.

*grumble*
posted by Alex404 at 11:43 AM on August 1 [30 favorites]

But how old is the ship captain?
posted by thelonius at 11:43 AM on August 1 [1 favorite]

OMFG YOU GUYS IS IT TRUE OR FALSE IDK MIND BLOWN THERE IS NO GOD

it's true and it's false. God has a sense of humor.
posted by philip-random at 11:46 AM on August 1

A +1 or a +16 to sjswitzer for pointing out the two types of multiplication.

It's perverse, perverse I tell you, to do the division first in 8 ÷ 2(2 + 2).

But with 8 ÷ 2 * (2 + 2), my computer-language instincts take over: ÷ then *.

You don't get the multiplication-by-juxtaposition thing in computer languages since you have multi-character variable names.
posted by zompist at 11:46 AM on August 1 [2 favorites]

Can we at least agree that the order of precedence between the bitwise and relational operators in C/C++/Java is wrong and Javascript fixed it?
posted by sjswitzer at 11:46 AM on August 1 [1 favorite]

Is it colder in the mountains, or in the summer?
posted by Token Meme at 11:47 AM on August 1 [7 favorites]

Laurel.
posted by sacrifix at 11:47 AM on August 1 [5 favorites]

This is where taking literature classes instead of the sciences really pays off. The answer is obviously 4, with the parenthetic expression just being a restatement of the claim in simplified form for other humanities majors who think division is a pain.
posted by gusottertrout at 11:48 AM on August 1 [26 favorites]

(puts on occasional compiler writer's hat)

OK it's P-E-DM-AS and B-O-DM-AS - addition/subtraction are evaluated left to right, so are division/multiplication
posted by mbo at 11:49 AM on August 1 [3 favorites]

God, this whole thing. So, over the years I've moved through stage one (arguing vociferously about the specific example raised), and through stage two (holding forth grumpily about the presentation of a poorly communicated math proposition as some kind of paradox), and am now occupying stage three (restfully contemplating the seeming inevitability of this making the circuit again). I'm not sure what stage four is and I'm interested and afraid to find out.

I wonder if anyone has charted the resurgences of this kind of thing on social media? Like, it's the kind of easy goofy argument chum (I'm not even against it, goofy arguing is fun) that click-chasing outlets are probably constantly repurposing as clickbait, but every once in a while it takes off, and the internet argues fervidly for a few days about arithmetic results and talks about order of operations and folks end up ultimately reiterating that using parentheses and sensible notation is the only real route away from this deliberately-constructed confusion. And then it's over. And then, at some point, it happens again. And again.

Is there a steady heartbeat? What's the internet's refractory period on these arithmetic paroxysms? Can we create a model to predict optimal timing for the next one to go viral? Is starting to take this question seriously a symptom of the onset of stage four?
posted by cortex at 11:49 AM on August 1 [22 favorites]

...no one can agree on an answer
I see lots of people agreeing. I'm even agreeing with most of them.
If you're expecting everyone to agree on anything, even the earth not being flat, it's going to be a long wait.
posted by MtDewd at 11:49 AM on August 1 [4 favorites]

I take it back, the answer is 10. Because I used the correct base.
posted by mrgoat at 11:50 AM on August 1 [10 favorites]

What pisses me off the most is that they could type ÷ but not × or ·. They're only being half fancypants. The fact that they didn't put an operator for the multiplication and relied on the implicit non-digits with no space is multiplication sorta lowers the precedence of the division below the multiplication. PHBBBBT.

On preview sjswitzer hit it while I was futzing around.
posted by zengargoyle at 11:51 AM on August 1 [1 favorite]

I'm not sure what stage four is and I'm interested and afraid to find out.

If I remember how these stories go, I believe we're supposed to get clonked on the head, at which point we will achieve enlightenment.
posted by echo target at 11:53 AM on August 1 [1 favorite]

As a bold, brave thinker who refused to click the article and learn the problem... I declare...…

BANKRUPTCY!!!!!!!!!
posted by OnTheLastCastle at 11:54 AM on August 1 [1 favorite]

I thought searching for "PEDMAS" would turn up more than one other MeFi thread, but, hey, here's one from 2013.
posted by cortex at 11:56 AM on August 1 [2 favorites]

I'm a programmer by trade, but I'm on team "add parentheses to disambiguate, because this is dumb to argue about otherwise".

I say "but" in the prior sentence because I know a lot of programmers will argue to the death about how because we know the order of operations in a given language you should never add parentheses when you know you'll get what you want without them. Meanwhile I'm over here in the real world where people are imperfect an have to switch contexts all the time and it's not worth prioritizing the efficiency of 2 fewer characters in a statement if it means there's a non-zero chance someone will misunderstand it when seeing it years later.
posted by tocts at 11:57 AM on August 1 [13 favorites]

If you think this is hilarious, try writing a fully deterministic C program.
posted by seanmpuckett at 12:00 PM on August 1 [5 favorites]

There should be a succinct, negatively-connotating term for these utterly pointless, inconsequential clickbait non-issues where every position is equally valid (or invalid) and serve only to inflame division and cause otherwise normal, level-headed individuals to waste time and emotional energy engaging in ironic "fights" where the ultimate loser is every living being and the ultimate winner is, inexplicably, advertisers.

I'd like to think we're better than this shit but maybe I'm just a fucking moron.
posted by glonous keming at 12:02 PM on August 1 [6 favorites]

About as interesting as arguing whether you spell /əˈpɑl·əˌdʒɑɪz/ apologize or apologise.
posted by mono blanco at 12:08 PM on August 1 [1 favorite]

It's just good-natured tongue-in-cheek sparring like dogs vs. cats, hotdogs are sandwiches?, and the rest of our periodic foodfight threads.
posted by sjswitzer at 12:09 PM on August 1 [1 favorite]

no one can agree on an answer

Is there anyone who uses math in any professional or graduate-level capacity who disagrees with the consensus that the answer is 16?
posted by qxntpqbbbqxl at 12:13 PM on August 1

New question: PEMDAS? BODMAS?
So, I just learned now that only Canada & New Zealand use BEDMAS, and the rest of the world use other terms? No wonder this is unclear.

Also... 1
posted by Laura in Canada at 12:18 PM on August 1 [4 favorites]

Is there anyone who uses math in any professional or graduate-level capacity who disagrees with the consensus that the answer is 16?

Well, yes.

Again, I ask what is "a / bc"?

To me it feels tortured to argue that it's "(a / b) × c".
posted by sjswitzer at 12:20 PM on August 1 [7 favorites]

This is exactly the kind of problem you (and by you, I mean me) edit out of 5th grade textbooks because both PEMDAS and BODMAS fail to adequately address when to perform implied multiplication when two values are right next to each other.
posted by 23skidoo at 12:22 PM on August 1 [4 favorites]

The rule I learned was that division and multiplication have equal precedence and, when you're talking about things with equal precedence, you evaluate from left to right. So, 16.

My question (which I don't know the answer to) is if there actually is a rule taught somewhere that multiplication comes before division (in which case the answer would be 1) or if this is just people misinterpreting the rule they were taught.

The PEMDAS vs BODMAS thing sort of seems like a red herring, because both in PEMDAS (which I learned) and in BODMAS (which I googled), it seems like Division/Multiplication and Addition/Subtraction are on the same level.
posted by Betelgeuse at 12:27 PM on August 1 [3 favorites]

This is exactly the kind of problem you (and by you, I mean me) edit out of 5th grade textbooks because both PEMDAS and BODMAS fail to adequately address when to perform implied multiplication when two values are right next to each other.

Why? Isn't multiplication multiplication whether or not that multiplication is implied or explicit?

multiplication
posted by Betelgeuse at 12:29 PM on August 1 [2 favorites]

Because precedence is a property of the notation and not of the operations themselves.
posted by sjswitzer at 12:30 PM on August 1 [6 favorites]

There is no room for philosophical argument, there is only the Compiler. And what the Compiler says, goes.

Implicit multiplication is a syntax error*. Division is prior to multiplication**.

* usually
** usually
*** usually

posted by klanawa at 12:31 PM on August 1 [1 favorite]

I keep seeing it as the multiplication that is implied by the bracket is part of the "bracket rule". You add the stuff inside the bracket, then multiply anything stuck to it. Then follow the rest of the order of operations.
posted by Laura in Canada at 12:32 PM on August 1 [7 favorites]

For instance you would not have the same question if it were written

a
---
bc

or even

a
------
b * c

In these cases it's clear that you group b*c and that is a property of the notation and not the operations themselves.
posted by sjswitzer at 12:33 PM on August 1 [3 favorites]

Well, yes.

Again, I ask what is "a / bc"?

To me it feels tortured to argue that it's "(a / b) × c".

The notation is purposely non-standard (which is why there is arguing about it), but following the order of operation rule, that's the correct interpretation.

I wonder if the arguing would persist if the implicit multiplication was made explicit.
posted by Betelgeuse at 12:36 PM on August 1 [1 favorite]

Betelgeuse said: Why?

Hey, if *you* want to not edit that kind of problem out of 5th grade textbooks, you do you, but the order of operations is supposed to make things easier for students, not harder. Solve a/bc for a = 8, b = 2, c = 4. That problem is just going to make parents call teachers because they disagree with an answer in the back of the book, and I don't want to do that to a 5th grade math teacher. It's just way easier to write problems that aren't ambiguous.
posted by 23skidoo at 12:37 PM on August 1 [4 favorites]

I keep seeing it as the multiplication that is implied by the bracket is part of the "bracket rule". You add the stuff inside the bracket, then multiply anything stuck to it. Then follow the rest of the order of operations.

Now that is interesting. Again, I wonder if this is a misinterpretation of the rule as taught or if this is actually the official convention in some places.
posted by Betelgeuse at 12:38 PM on August 1

This is why I use an RPN calculator. -Doomsey

I just recently got new batteries for my mother's 1989s era HP12c after seeing the head of my office using his. I need the RPN for dummies book though so I can confuse and baffle my coworkers instead of just doing simple math. Sadly, no one has asked to borrow it yet.

(Also, I thought Reverse Polish Notation was just something my dad made up rather than a real term)
posted by vespabelle at 12:43 PM on August 1 [4 favorites]

Historically, people were not rigorous with notation and originally--and necessarily--made it up as they went along. The "rules" were written to explain what had eventually become common, though not entirely consistent, usage. Whether the rules captured that satisfactorily is another thing entirely. There is wide agreement that the rules around juxtaposition are unspecified and that avoiding it is the best practice.

IOW, this is descriptive versus prescriptive grammar all over again.

ETA: ... avoiding ambiguous usages of it is the best practice.
posted by sjswitzer at 12:45 PM on August 1 [4 favorites]

came here hoping that someone else had already mentioned rpn, left satisfied.
posted by Reclusive Novelist Thomas Pynchon at 12:48 PM on August 1 [1 favorite]

@Betelgeuse: I believe that is how I was taught (but has been more than a few years). For me, if the multiplication implied by the bracket was made implicit that answer would be 16. As shown, I see 1.
posted by Laura in Canada at 12:50 PM on August 1 [1 favorite]

came here hoping that someone else had already mentioned rpn, left satisfied.

The k3w1 k1dz use point-free notation now.
posted by sjswitzer at 12:51 PM on August 1

also: it's 16, it's white and gold, when you read it out loud it sounds like yanny, and i don't believe anyone who says anything else.
posted by Reclusive Novelist Thomas Pynchon at 12:52 PM on August 1

I got some PEBCAK in my PEMDAS.
posted by kimota at 12:54 PM on August 1 [5 favorites]

I only have high school maths but I've never heard of a rule that implicit notation takes precedence over explicit notation. In fact, I've been taught explicitly that ab is exactly the same as a*b.
posted by patrick54 at 12:55 PM on August 1 [1 favorite]

I am curious in terms of programmers vs non-programmers vs math geeks (who may or may not be programmers), and uh...none of the above people...

What's the general rate. I imagine the super thoroughly rigorous do the simple escape of:
"Well the definition is ambiguous therefore it's not a valid question" and demand a non-ambiguous refinement/clarification.

I imagine those who do general coding but not as a job, have a certain preference/perception of how things are done. I count my self in the "1" camp. To me, I think it comes from something like the above mentioned that the () is a multiplication and thus binds to that operation (including the stuff you're doing inside) so you complete that last multiplication before moving to the division.

I think this comes from my own appreciation for LISP style syntax (which, is just Polish Notation), even though I don't code lisp (in fact when trying to code it in LISP syntax as written/as I interpret it, I can't get it to run, so I don't know lisp that well). But it seems to me, we're just moving right to left here something like:

(/ 8 (* 2 (+ 2 2)))

LISPers am I doing that right kinda?

And what would the 16 answer look like in LISP?
posted by symbioid at 12:57 PM on August 1

I imagine it would be (* (/ 8 2) (+ 2 2)) or so?
posted by symbioid at 12:59 PM on August 1

it's 16 and it wears pants like this:
    1       66  111      6 1  1      6  below  this  line ---1------666------------    1      6  6  is pants   111111    666              
posted by Reclusive Novelist Thomas Pynchon at 1:01 PM on August 1 [8 favorites]

okay half the reason i typed that up was as a cruel prank for people who try to quote it
posted by Reclusive Novelist Thomas Pynchon at 1:05 PM on August 1 [1 favorite]

I'll just quote a snippet: "666" :)
posted by sjswitzer at 1:09 PM on August 1

hello PEMDAS my old friend / I've come to talk with you again
posted by BungaDunga at 1:12 PM on August 1 [2 favorites]

you haven't lived until you've programmed in a language without operator precedence which just blindly evaluates left to right unless you put parentheses in
posted by BungaDunga at 1:15 PM on August 1 [3 favorites]

The actual order of operations is groups as follows:

1. P
2. E
3. (MD)
4. (AS)

Each of the steps (1 through 4 ) is executed serially and from left to right.

Therefore, it becomes
8 ÷ 2(2+2)

8 ÷ 2(4)

4 ÷ (4)

1
posted by ianhorse at 1:15 PM on August 1 [2 favorites]

I’m going to take a hardline stance on this and advocate for never using subtraction or division operators at all, only unary inversion operators as in -x and x^-1. Real numbers along with those two operations are a commutative algebra, and we shouldn’t taint their essence with operations that aren’t.
posted by invitapriore at 1:16 PM on August 1 [9 favorites]

I'll just quote a snippet: "666" :)

yeah the way to put multiple spaces in a row in an html document is to use the &hssp; html entity, which stands for "hail satan space."
posted by Reclusive Novelist Thomas Pynchon at 1:21 PM on August 1 [2 favorites]

Anyway PEMDAS is what it is at least mostly so you write down polynomials without parentheses.
posted by BungaDunga at 1:22 PM on August 1 [2 favorites]

8 ÷ 2(2 +2) is dividing the internet, and no one can agree on an answer.

What we need now, more than ever, is coming together. 8 ÷ 2(-1 +2) can unite the internet. Text PEMDAS to 33030 to learn more about 8 ÷ 2(-1 +2).
posted by 23skidoo at 1:25 PM on August 1 [7 favorites]

It's about the journey, not the destination.

But, the destination is 1.
posted by GoldenEel at 1:26 PM on August 1 [3 favorites]

James T. Kirk's day just got a lot easier. ILLOGICAL! ILLOGICAL! 16! 16! 1! KABOOM!!!
posted by justsomebodythatyouusedtoknow at 1:28 PM on August 1 [2 favorites]

That's numberwang?
posted by youknowwhatpart at 1:29 PM on August 1 [14 favorites]

Ever since I was taught it, I've thought order of operations was a stupid idea. The problem. as here, is that it often doesn't properly or unambiguously express distribution or association.

And that similarly the singular division and multiplication signs are terrible ideas, for this same reason. At least in chemistry, we're honest that we're teaching successive approximations. Math needs to lose the "order of operations" nonsense from the start to be really avoid this kind of ambiguity.
posted by bonehead at 1:38 PM on August 1

> I'm a programmer by trade, but I'm on team "add parentheses to disambiguate, because this is dumb to argue about otherwise".

Also, explicit expressions makes code more accessible to people with dyslexia, dyscalculia or vision problems.
posted by ardgedee at 1:38 PM on August 1 [1 favorite]

I tried it in APL, and apparently the answer is 4 2         (That's not 42)
Everyone knows you do operators right-to-left anyway.
posted by MtDewd at 1:41 PM on August 1 [2 favorites]

Does anyone agree with me that the phrase "no one can agree on an answer" is dumber than the ambiguous math notation?
posted by straight at 1:41 PM on August 1 [4 favorites]

This is why I use an RPN calculator.

Now you just have to figure out which camp you belong in with regards to evaluating the expression under discussion, so you can input that interpretation into your unambiguous calculator.

You can't escape.

(It's 1 as written, but 16 if you put an actual multiplication operator before the parentheses.)
posted by Dysk at 1:48 PM on August 1 [5 favorites]

In Wingdings font the equation becomes (and you have to go with me here, as I can't type Wingdings in this box so i'm just going to describe it).... computer mouse, left down arrow, document, old phone, document, mail, document, phone call.

I interpret this as "move your mouse off this window and get back to the work, email, calls you are supposed to be doing." Good enough.
posted by inflatablekiwi at 1:57 PM on August 1 [3 favorites]

According to my calculations the answer is 80085.
posted by Pyrogenesis at 2:09 PM on August 1 [3 favorites]

Anyway PEMDAS is what it is at least mostly so you write down polynomials without parentheses.

In effect, yes! Mathematicians working on polynomial problems had worked out a notation that was efficient for them, and that notation eventually became common. It was then a problem for educators to explain it and they did it... variously. And they really didn't give much thought to the juxtaposition notation vis-a-vis / which is, in any case, rare in proper mathematical notation. Handwritten and typeset math uses the fraction bar (-----).

Mathematical notation never uses * (as multiplication) and rarely uses × or ÷ or ∕.

Limitations of linear text forced programming languages to use / and, for lack of a × and because of, as pointed out earlier, the use of multi-character variable names, a * as a sad compromise.

Unicode is now ubiquitous and we could be using symbols like ÷ (which itself is just a reference to the fraction bar) and × in programming languages but it has not caught on, because it's inconvenient to type and because it's rarely used in math anyway.

What I posit here is that, when given typographical tools to express math as it is written, "/" and juxtaposition just don't come up in the same expressions, so it hasn't been an issue to solve... outside of fighty internet things (and very unfortunate grammar school tests).

Another thing I want to highlight here is the issue of "order of operations," grammar, and notation. Order of operations is a grammatical concept of precedence. However it is a sketchy one because it confuses the nature of the operation itself with the grammar used to express it. Grammatically, precedence attaches to the symbols * (or ×) and / (or ÷) and not to the operations multiplication and division. There is nothing about multiplication as an operation that is more tightly-bound than addition as an operation. It's just a notational convenience, realized in the grammatical rule that multiplication (really × or juxtaposition), comes before addition (really +). BUT... grammar is usually formalized as a set of rules over a linear sequence of symbols and mathematical notation is in fact a two-dimensional arrangement of symbols. There are systems to express those notations linearly (LaTeX, etc.), but the actual notations are not linear. It's possible to devise "grammars" for 2D notations and in formal math these notations are precisely defined. But that kind of formalism is of no use in primary education.

So we are left with this mess where "bc" as written clearly expresses the intention to multiply b and c but the rules, which are at two removes from the actual notation (first the 2D actual notation, second the implied but often misleading linearization of that notation into a string, and finally the rules which, sadly are expressed in terms of the operations and not the notations or even the 1D grammar) fail to capture what the notation is meant to express.

It's a sad state of affairs, but it doesn't plague actual mathematicians, who have few practical confusions about their notations. And it doesn't much affect programmers who generally, but not infallibly, also have few practical confusions about what their (different) notations mean (C++ programmers excluded). The brunt of this falls on educators and their unfortunate students who have to learn just enough to make sense of these notations.
posted by sjswitzer at 2:25 PM on August 1 [10 favorites]

(It's 1 as written, but 16 if you put an actual multiplication operator before the parentheses.)

Only if you take implicit multiplication to have a higher precedence than explicit multiplication, which PEMDAS doesn't cover.
posted by BungaDunga at 2:26 PM on August 1

Weird, I got Joe 30330
posted by oulipian at 2:30 PM on August 1 [7 favorites]

weird I keep getting 5318008
posted by nikaspark at 2:34 PM on August 1 [12 favorites]

÷ (which itself is just a reference to the fraction bar)

I'm not convinced of that one myself. The obelus was a typographical symbol that was only borrowed to mean division in the 17th century and was sometimes used for subtraction instead (even more confusing!).
posted by BungaDunga at 2:34 PM on August 1 [2 favorites]

consider: PEDMSA
posted by BungaDunga at 2:36 PM on August 1 [1 favorite]

It's been
8 ÷ 2(2+2) week since you looked at me(fi)
posted by rather be jorting at 2:41 PM on August 1 [13 favorites]

Anyway if you don't want to invoke polynomials, the best argument for multiplication coming first is that "three dozen eggs and five more" and "five eggs and three dozen more" are both the same number of eggs.

(or "1 penny and eight schillings" and vice versa is the same number of pennies whichever order you say it in)
posted by BungaDunga at 2:41 PM on August 1 [1 favorite]

PEDMSA

Parentheses, Exponents, Division Means Subtraction, Addition
posted by 23skidoo at 2:44 PM on August 1 [2 favorites]

A lovely grammatical defense of juxtaposition precedence :)
posted by sjswitzer at 2:45 PM on August 1 [1 favorite]

I got Yanni
posted by jquinby at 2:46 PM on August 1 [1 favorite]

I asked my computer and it said TypeError: 'int' object is not callable which probably means "that question is so stupid we're not even allowed to say the word I'd like to call it."
posted by sfenders at 2:50 PM on August 1 [1 favorite]

"three dozen eggs and five more" and "five eggs and three dozen more" are both the same number of eggs

Yes, though "the product of 3 and 12 plus 5" has some degree of ambiguity, while "the sum of 5 and the product of 3 and 12" does not.
posted by 23skidoo at 2:52 PM on August 1

Isn't the correct answer "Don't respond to intentionally ambiguous social media posts because most of the time they're padding their engagement stats in the hope of monetizing an otherwise worthless page or account?"

I am enjoying getting the discussion getting into the weeds of why it's ambiguous in the first place though. Far more enjoyable than a string of 1's and 16's with the occasional snarky "Didn't you learn BOMDAS in high school!" comment.
posted by TwoWordReview at 3:14 PM on August 1 [2 favorites]

>>> 7/2*(2+2)
12
posted by clawsoon at 3:14 PM on August 1 [1 favorite]

I didn't even know there was such a thing as BODMAS. I clearly remember Mr. Toth in 8th grade algebra pounding Please Excuse My Dear Aunt Sally into my head. And then there was Mr. Bernath in 10th grade algebra 2 pounding the quadratic formula. There is very little I remember from math in school, but those 2 I will never forget.
posted by kathrynm at 3:59 PM on August 1

But is the equation black and blue, or is it white and gold?
posted by Quackles at 4:11 PM on August 1

I only have high school maths but I've never heard of a rule that implicit notation takes precedence over explicit notation. In fact, I've been taught explicitly that ab is exactly the same as a*b.

a(b+c) = (ab + ac)

and that doesn't change if you add multiplication/division or addition/subtraction terms to the equation so

x ÷ a(b+c) = x ÷ (ab + ac)

This isn't the case for

a*(b+c)

So in this case the answer, as I was taught it not a mis-remembering, is 1.

Even knowing the theory behind the 16 answer I'm having trouble parsing the equation that way as a whole. Likewrapping your head around a purple orange or a blue stop sign.
posted by Mitheral at 4:11 PM on August 1 [1 favorite]

Canadian here, and all of yall with your BODMASes and PEMDASes are wrong! It's BEDMAS. WTF even is an "order"?
posted by Dr. Send at 4:22 PM on August 1 [2 favorites]

Is there anyone who uses math in any professional or graduate-level capacity who disagrees with the consensus that the answer is 16?

In my case, it would depend in a number of factors including:
* Which sub-field of math are we working in? Different ones have different notational conventions.
* Where did the person who wrote the expression receive their mathematical training? In North America, we're often used to reading things left to right, but some other places have different conventions - at least at more advanced levels (linear and abstract algebra), though sometimes this carries back to elementary arithmetic too.
posted by eviemath at 4:23 PM on August 1 [2 favorites]

Nobody over the age of seven should ever use "÷", and TBH I think it's pretty dubious to teach this symbol even to children.
posted by Pyry at 4:29 PM on August 1 [7 favorites]

^ this
posted by sjswitzer at 4:31 PM on August 1 [3 favorites]

Count me with those saying mathematical notation is convention, just as grammar is. This expression is slang, and will be parsed differently (or unparseable) depending on your speech community. Claiming that this has One True Meaning is as silly, and incorrect, as claiming that "caddy-corner" is right and "kitty-corner" is wrong. Standard math notation should avoid constructions like this as ambiguous, in the same way that standard English should adhere to good spelling even if u cud reed it rite aneewae. 1 and 16 both make sense depending on which vernacular you use, as does rejecting the expression as ambiguous.
posted by biogeo at 4:36 PM on August 1 [4 favorites]

And division should always be written as exp(log(a)-log(b)). Also the only logarithm is the natural logarithm.
posted by biogeo at 4:38 PM on August 1 [2 favorites]

Pyry is correct. ÷ (or /) and juxtaposition multiplication do not belong in the same expression. It is a mixing of registers of language. There's no surprise the "rules" don't cover it because it would simply never occur to a practitioner to consider the problem.

There has to be a pedagogical path from counting on fingers to, say, algebra. In this path, certain things have to be introduced and discarded. ÷ is prime among them, as is its typographical doppelganger /.
posted by sjswitzer at 4:38 PM on August 1 [1 favorite]

Isn't multiplication multiplication whether or not that multiplication is implied or explicit?

I scratched around on the internet and found this history of the order of operations from 2000 on Ask Dr. Math (where math students and math teachers used to be able to ask questions about math to people studying math in college and get stuff explained- no longer accepting questions, but the archive is full of great questions and answers). A pull quote:

4. I suspect that the concept, and especially the term "order of operations" and the "PEMDAS/BEDMAS" mnemonics, was formalized only in this century, or at least in the late 1800s, with the growth of the textbook industry. I think it has been more important to text authors than to mathematicians, who have just informally agreed without needing to state anything officially.

5. There is still some development in this area, as we frequently hear from students and teachers confused by texts that either teach or imply that implicit multiplication (2x) takes precedence over explicit multiplication and division (2*x, 2/x) in expressions such as a/2b, which they would take as a/(2b), contrary to the generally accepted rules. The idea of adding new rules like this implies that the conventions are not yet completely stable; the situation is not all that different from the 1600s.

posted by 23skidoo at 4:44 PM on August 1 [3 favorites]

Didn't even know BODMAS existed. Sounds like a Dr. Who villain.
posted by Ray Walston, Luck Dragon at 4:45 PM on August 1 [1 favorite]

So much for "there are no stupid questions"
posted by ckape at 4:46 PM on August 1 [5 favorites]

I maintain that this is unsolvable until we determine what the function 2() does.
posted by NMcCoy at 4:58 PM on August 1 [3 favorites]

I feel like an FPP on the history of mathematical notation would be useful here. We tend to take it as a given and defined by rules. Nothing could be further from the truth. It's a contingent result of a lot of sequential and parallel effort and the result is... pretty good! But it's arbitrary and settled only in the consensus view (again, at the practitioner level). Also, math notation as we know it is relatively recent. Earlier math students would have to accommodate numerous idiosyncratic and conventional notations as a matter of course.

(Even today, the the decimal point notation is not internationally standardized (. or ,), nor the notation for grouping digits in clumps. Don't even get me started on what a "billion" means. The point is, there are a lot of more fundamental things there are no firm "rules" for.)

People who have to teach it at various levels have a difficult problem. They need to come up with rules for an unruly situation. No, actually, a progression of rules, some of which leave the other rules behind as stepping stones. And this is even apart from what the point of early mathematics education even is... is it basic understanding, facility in everyday uses of math, or a stepping stone to higher math?
posted by sjswitzer at 5:00 PM on August 1 [4 favorites]

Count me with those saying mathematical notation is convention, just as grammar is.

Sure. So is what side of the road to drive on. That doesn't mean you can just pick what side you feel like using today.
posted by thelonius at 5:06 PM on August 1 [2 favorites]

And as one moves to increasingly more specialized domains, it remains true still that students and practitioners "have to accommodate numerous idiosyncratic and conventional notations as a matter of course", as sjswitzer says. For example, what is the meaning of an expression like xy? Is this x raised to the power y, or is it x indexed with y? In the Ricci calculus for tensors, used extensively in modern relativity, you need two different kinds of index, and xy and xy both mean indexing x with y, but the first case is contravariant and the second is covariant. Because powers and polynomials are comparatively infrequent when solving problems in this domain, mathematicians and physicists have found it more convenient to abandon the convention that a small superscript means exponentiation and adopt instead the Ricci calculus convention. Once you know what domain you're working in and what conventions are in use, there's no ambiguity.
posted by biogeo at 5:10 PM on August 1 [3 favorites]

Sure. So is what side of the road to drive on. That doesn't mean you can just pick what side you feel like using today.

On the other hand, which side of the road you use today is going to depend entirely on where you are and what the general local consensus is about which side of the road everybody is using. It's not a natural law of the universe, it's a convention of practice, and it varies from one context to the next.

The baseline takeaway of this link is, in this iffy analogy, something like "don't build roads that mix signals about which side of the road folks are supposed to be driving on, and then act shocked by head-on collisions".
posted by cortex at 5:13 PM on August 1 [4 favorites]

Ctrl + f 'polish notation'
2 results found

I knew y'all were my kind of nerds.
posted by Mayor West at 5:15 PM on August 1 [1 favorite]

Sure. So is what side of the road to drive on. That doesn't mean you can just pick what side you feel like using today.

But this isn't really picking a side of the road to drive on. I mean, if you're talking about something like equations being used to get rockets to the moon or something, then yes, you need rock-solid, unambiguous interpretation, and in that case the right answer would be "this expression is too ambiguous to be useful, write it a different way." But for most purposes this is like picking a side of the sidewalk to pass by someone coming the other way. Yeah, it's more efficient if everyone agrees, but if I'm walking a relatively sparsely populated sidewalk and someone coming the other way picks the same side as I do to try to move to to pass each other, the worst that happens is we do an awkward dance for a couple seconds until we resolve the confusion.

Strict rules about what side of the road to drive on are important because:

1. Drivers can't readily communicate with each other in real time to negotiate this
2. Even if they could, the costs of this negotiation would be very high (traffic jams)
3. The costs of failing to reach an agreement are very high (collision)

Situations governing grammatical rules and mathematical syntax sometimes meet these conditions (e.g., legal documents, computer programs), but not always (e.g, chatting with friends, solving a basic physics problem), so I don't think the "side of the road you drive on" analogy is necessarily a good one.
posted by biogeo at 5:21 PM on August 1

The only real argument worth having is whether to use prefix or postfix notation. On the one hand, subject-object-verb word order is the most common among the world's languages. On the other hand, verb-subject-object languages include Welsh, which has that awesome "Ll" phoneme. I'm prepared to entertain the idea that we could adopt some kind of case system for mathematical expressions, like Latin or Sanskrit, instead of prefix or postfix notation, but then we need a new notation for case markers on numbers and variables. But that's the only context in which I'm prepared to let people keep their silly infix notation.
posted by biogeo at 5:31 PM on August 1

8 ÷ 2(2+2) looks like trolling, but there are less flashy examples of this ambiguity that even good communicators are susceptible to. Does 2/3x in an email mean ⅔x or 2/(3x)?

We expect that any language rich enough to have poetry will also have structural ambiguity (e.g., Man shot by Cheney to make statement -- N. Y. Times). These slips are intrinsically hard to catch; they appear like imps, not only punishing bad writers who don't follow the rules, but mocking good writers who do. Expecting mathematics not to have this problem betrays low expectations for how much mathematicians should want to be able to say.
posted by aws17576 at 5:33 PM on August 1 [5 favorites]

A love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit.
Come, let us hasten to a higher plane,
Their indices bedecked from one to n,
Commingled in an endless Markov chain!

Come, every frustum longs to be a cone,
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze:
It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space
Let superscripts and subscripts go their ways.
Our asymptotes no longer out of phase,
We shall encounter, counting, face to face.

Thou'lt tell me all the constants of thy love;
And so we two shall all love's lemmas prove,
And in our bound partition never part.

For what did Cauchy know, or Christoffel,
Or Fourier, or any Boole or Euler,
Wielding their compasses, their pens and rulers,
Of thy supernal sinusoidal spell?

Cancel me not -- for what then shall remain?
Abscissas, some mantissas, modules, modes,
A root or two, a torus and a node:
The inverse of my verse, a null domain.

Ellipse of bliss, converse, O lips divine!
The product of our scalars is defined!
Cyberiad draws nigh, and the skew mind
cuts capers like a happy haversine.

I see the eigenvalue in thine eye,
I hear the tender tensor in thy sigh.
Bernoulli would have been content to die,
Had he but known such a squared cosine 2 phi!
posted by biogeo at 5:41 PM on August 1 [4 favorites]

Does 2/3x in an email mean ⅔x or 2/(3x)?

That's a really great example! You could not know without context. However, even whitespace would give a sympathetic reader a clue as to the meaning: "2/3 x" vs. "2 / 3x". There are no "rules" for this, but the whitespace would disambiguate it for most readers.

And also, the only reason there's an issue is that it's hard (and lossy) to translate conventional 2D notation to 1D ASCII.
posted by sjswitzer at 5:46 PM on August 1 [4 favorites]

Returning to the pedagogy of it, I want to contrast two approaches:

1 - Can you apply some rules by rote?
2 - Can you understand what was meant by this expression?

Which one would you prefer was taught in school?

(Granted, this particular expression was ignorantly formed, and it's unfair for students to have to deal with it.)
posted by sjswitzer at 5:55 PM on August 1

Their indices bedecked from one to n

Their indices bedecked from zero to n minus one :)
posted by sjswitzer at 6:00 PM on August 1 [2 favorites]

Can you understand what was meant by this expression?

I mean, that's the thing really- the whole point of teaching the order of operations to schoolkids is not to simplify numeric expressions to single number, the point of teaching the order of operations is 1) so that when kids start writing algebraic equations that correspond to a specific situation, they're better equipped to write an equation that actually relates to the situation they're writing the equation for, and 2) when kids start solving equations, (imo) understanding the order of operations can be helpful because you (can) kind of use the order of operations in reverse to solve simple algebraic equations like 3x + 12 = 14. Like, you *could* start the problem by using division to divide both sides of the equation by 3, but it's (likely) much easier (for kids) to start by using subtraction to subtract 12 from both sides of the equation.
posted by 23skidoo at 6:11 PM on August 1 [2 favorites]

Just to be double-contrarian, I'll say
8 / 2(2 + 2)
= 8 / 4 + 4
= 2 + 4
= 6
posted by ctmf at 6:33 PM on August 1 [4 favorites]

At least we've solved this one:

['🐔' , '🥚'].sort() = ['🐔' , '🥚']
['🥚' , '🐔'].sort() = ['🐔' , '🥚']
posted by bendy at 6:52 PM on August 1 [5 favorites]

If it's ambiguous clearly it needs to be phrased better. If it should only have one answer and it has two, then again, it needs to be phrased better.

It's stupidly presented, that's all. Add another set of brackets.
posted by adam hominem at 8:09 PM on August 1 [1 favorite]

At least the equation wasn’t apple plus banana plus sandwich = ? Which is the usual equation dividing the internet. Also curious which bucket fills first.
posted by snofoam at 8:24 PM on August 1

Didn't even know BODMAS existed. Sounds like a Dr. Who villain.

Or an obscure English holiday. I think it’s a month after Michaelmas.
posted by delicious-luncheon at 8:54 PM on August 1 [2 favorites]

apple plus banana plus sandwich = ?

posted by fings at 9:00 PM on August 1

At least the equation wasn’t apple plus banana plus sandwich = ?

ha, a mathy FB friend of mine juuuuust posted something which on the surface appears to be one of those apple + banana + sandwich problems, but it actually is a simple-to-understand but hard-AF to solve monster of a math problem that's pretty much unsolveable for almost everyone except for number theorists.
posted by 23skidoo at 9:20 PM on August 1 [4 favorites]

There's something special about learning rules in school when you are 12 that make people super invested in them. It's like you're old enough to remember them but too young to wonder with a sixth grade textbook writer is really that great an authority.

Best you can say about this one is it's probably distracting the participants from arguing about prepositions at the end of sentences, the singular "they," or the proper spelling of aluminum.

Meanwhile an actually funny order of operations puzzle. (I guarantee you the given answer is correct.)
posted by mark k at 9:20 PM on August 1 [10 favorites]

Okay but the answer is 1, right?
posted by Mchelly at 9:51 PM on August 1

Yes
posted by sjswitzer at 9:55 PM on August 1

(internal screaming)
posted by Kutsuwamushi at 11:03 PM on August 1

posted by sjswitzer at 11:20 PM on August 1

Weirdly, it's actually -1/12.
posted by biogeo at 11:49 PM on August 1 [7 favorites]

As a millennial with pre-new math education, my conclusive answer is 1.
Wait, is this a re-definition of "New Math"? I thought it was the educational system of starting from underlying concepts of base systems that Tom Lehrer lampooned in the early 1960s.
posted by rum-soaked space hobo at 2:14 AM on August 2 [3 favorites]

Currently having math homework flashbacks where I could, with a great amount of effort, fail to get the same answer -- twice.

/rubs temples
posted by redrawturtle at 3:02 AM on August 2

The bell pushed by the wind rang.
posted by es_de_bah at 3:45 AM on August 2

Returning to the pedagogy of it, I want to contrast two approaches:

1 - Can you apply some rules by rote?
2 - Can you understand what was meant by this expression?

Which one would you prefer was taught in school?

Given that 1 seems to lead to really pointless internet arguments where people have very strong opinions based on completely arbitrary rules that they (in some cases - not saying this applies to fellow mefites) have no idea the purpose or history of, over expressions that are just poor grammar, would never appear in practice, have no meaning, and are just generally irrelevant...

I think I'll go with door #2.
posted by eviemath at 5:26 AM on August 2

Okay, but 8 eggs ÷ 2(2+2) = how many deviled eggs halves?
posted by eponym at 6:04 AM on August 2 [3 favorites]

Is the internet a number???

I didn't think it was, so how can it be divided??

In all my years of doing maths AND writing programs, I would say the answer is one, but the expression needs clarifying and if I was asked to verify it as code I would throw it back and ask that the expression be clarified with extra brackets.

Any decent programmer would not accept an expression that is open to interpretation or misunderstanding.
posted by Burn_IT at 8:24 AM on August 2

Adding on to this as a math teacher:

I think most math educators I know are aware that the order of operations we teacher is, of course, a convention that doesn't have a lot of mathematical value in and of itself. Other topics that you could describe that way include:
• base 10 notation
• using ordinary Roman letters to represent variables, rather than either words or other symbols
• having the x-axis point right and the y-axis point up
• focusing almost exclusively (at least at first) on linear (or maybe quadratic) functions
• deciding that zero, negative numbers, decimal and other rational fractions, and a select group of irrational numbers are worth knowing, but imaginary/complex numbers mostly can be ignored
• ...
But also if we just ignored the conventions of mathematics and of mathematics education, our students would flounder both on standardized tests and in their future mathematics career. There is value in learning the conventions of order of operations, just as much as there is value in learning how to use punctuation and spelling in a standard way, even if there's no reason beyond convention to facilitate clear communication.

If you want a (somewhat) better order of operations question, consider this one from the 2017 high school (10th-grade) standardized test in my state (which is a graduation requirement).
posted by thegears at 8:36 AM on August 2

The problem is that the grammars of order of operations don't have unambiguous interpretations, becasue a) there are multiple versions of them and b) they themselves are less than perfect interpretations of a more fluid and (frankly) intuitive notation which expresses multiplication and division positionally rather than symbolically.

This kind of question is ambiguous because of both factors. It's playing on the order or operations grammar issues, but, also cheating by partially using the positional placement and omitting the multiplication symbol. Personally, I speculate that it's designed to lead one used to a "higher math" positional notation to one result (1) while leading a person who is only used to order of operations to another (32). Since the first is largely taught intuitively, by absorption in math and related disciplines like physics, and the second formally, with textbooks and everything, it's a gotcha designed to make the "old math" PEDMAS person feel like the so-called "experts" can't even do basic arithmetic.

So climate change is obviously a hoax and homeschoolers rule.
posted by bonehead at 10:53 AM on August 2 [1 favorite]

I'm not seeing the problem. As far as I'm concerned, the expression 8 ÷ 2(2+2) is completely unambiguous and equal to zero -- because in my native language, Danish, the symbol ÷ has traditionally been used for subtraction, not division. Another reason not to use that symbol, I suppose.
posted by Tau Wedel at 12:12 PM on August 2 [4 favorites]

When I'm writing expressions in which it's convenient to use this kind of structure for my own personal use on my own personal bits of paper, I use the convention that implied multiplication works like the function application it resembles, so e.g. 2(3 + 4) implies enclosing parentheses as well as an infix multiply operator, making it equivalent to (2 × (3 + 4)).

When I read it in somebody else's text, it's usually easy enough to figure out which way they're doing it by context.

Arguing about which method is "correct" is as pointless as arguing about whether my little fuzzy friend Alfie is really a dog or wirklich ein Hund. As long as you're consistent it's all fine.

Also, APL and right-to-left evaluation ftw.

Oh yeah, internet? Well check this shit out:

This statement is false

OMFG YOU GUYS IS IT TRUE OR FALSE IDK MIND BLOWN THERE IS NO GOD

That's cool and all, but check this shit out:

This statement is true

OMFG YOU GUYS IS IT TRUE OR FALSE IDK MIND BLOWN THERE IS NO GOD
posted by flabdablet at 12:45 PM on August 2

Okay but the answer is 1, right?

0.9999999999999999
posted by bonje at 1:20 PM on August 2 [4 favorites]

I think most math educators I know are aware that the order of operations we teacher is, of course, a convention that doesn't have a lot of mathematical value in and of itself.

In this case though it seems like pre-college math educators have invented a convention that bears only a loose relationship to how math is actually written by practitioners. And worse, they seem to teach the idea that mathematical notation is universal and fixed, whereas in practice papers invent or use unconventional notation all the time, especially in newer subdomains which have had less time to settle on notation.

Also, standardized tests are about the only place where you will be given a bare expression stripped of all context and expected to interpret it: real scientific papers aren't (usually) just lists of equations you are meant to solve like puzzles.

(Finally because I can't resist commenting on it: typically instead of "1/2 x" you would see "x/2", which is both unambiguous and shorter [even in latex: "\frac{1}{2}x" is harder to read than "\frac{x}{2}"])
posted by Pyry at 1:54 PM on August 2

In this case though it seems like pre-college math educators have invented a convention that bears only a loose relationship to how math is actually written by practitioners. And worse, they seem to teach the idea that mathematical notation is universal and fixed, whereas in practice papers invent or use unconventional notation all the time, especially in newer subdomains which have had less time to settle on notation.

hahahahhahaha, you're seriously overestimating how much control pre-college math educators have over what they get to teach their math students.
posted by 23skidoo at 3:05 PM on August 2 [1 favorite]

Substitute in "pre-college maths textbook/syllabus authors" if it helps you understand the point.
posted by Dysk at 4:11 PM on August 2

"Have you seen this?" asked Edgar.

"It's very clever," added his twin Georg.

They sat either side of me and showed me the equation.

"But this is stupid," I told them. "It's not a puzzle. It's not a paradox. It's just deliberately annoying ambiguity."

Edgar leaned in from my left. Georg leaned in from my right. Suddenly my vision swam. The twins disappeared, and in their place sat a single white vase.

"Oh," said the vase sadly. "We rather like it."
posted by davidwitteveen at 5:01 PM on August 2 [4 favorites]

Substitute in "pre-college maths textbook/syllabus authors" if it helps you understand the point.

I write and edit pre-college math textbooks, if it helps you value my opinion.
posted by 23skidoo at 5:03 PM on August 2 [2 favorites]

Like, textbook editors and math educators don't get to decide what kids get to learn in a math class, bureaucrats who may or may not understand math well do.
posted by 23skidoo at 5:06 PM on August 2 [1 favorite]

Yeah but is .999...= 8 ÷ 2(2+2)
posted by BungaDunga at 5:59 PM on August 2

Like, textbook editors and math educators don't get to decide what kids get to learn in a math class, bureaucrats who may or may not understand math well do.

So yeah, syllabus authors.
posted by Dysk at 7:36 PM on August 2

(And that's for the States, I take it - who exactly decides what the kids learn varies pretty dramatically by country, school system, and individual school. So yeah, it's syllabus authors in the US, but it can easily be textbook authors, editors, or publishers in other systems, where the syllabus is less tightly defined for those producing textbooks, and so instead ends up being decided by them for their textbooks, and which particular textbooks by individual school groups, schools, or departments.)
posted by Dysk at 7:40 PM on August 2

You seem to be playing some weird semantic game where you just want to tell me I'm wrong, so maybe just drop it?
posted by 23skidoo at 7:45 PM on August 2

No, I'm pointing out that in some systems, it is educators and textbook authors, and in others, like the US, it's syllabus authors. The point is the same - people involved in pre-college maths education have defined a set of rules that at best approximate the conventions of maths as actually practised. The point you were rubbishing is perfectly cromulent, even if "educators" is perhaps at the wrong level for the US system, and should instead read "education bureaucrats" for that context instead.
posted by Dysk at 7:49 PM on August 2

And like, I have family who have, in their role as head teacher of a free school, been in charge of selecting syllabuses and textbooks for their school. This isn't some gotcha, this is how things actually operate in other parts of the world.
posted by Dysk at 7:52 PM on August 2

[be nice]
posted by Eyebrows McGee (staff) at 8:05 PM on August 2 [1 favorite]

The point you were rubbishing is perfectly cromulent

Disagree.
posted by 23skidoo at 8:11 PM on August 2

Dysk, idk if it's just me and my U.S.-centric way of thinking, but I'm confused by your use of the phrase "syllabus authors" as if it's a well-known term in the English language that someone in the U.S. might or ought to be aware of, could you elaborate?

I've never heard the combination of "syllabus" and "author" until this particular thread and I can't find elaboration when I search The Guardian or Google for "syllabus author" as a phrase. The primary association I (as a lifelong American) have with the word "syllabus" relates very little to the notion of authorship and is more along the lines of "college professor recycling last year's word .doc with the dates changed in the hopes that this semester will finally be the one where the class will always be on schedule." The notion of being a syllabus author, as if writing a syllabus is like writing an article or book or longform online screed, doesn't compute for me at all. How are you defining syllabus in the context of math education?

Additionally, I live in a state twice the size of the UK's entire land mass, where pre-university education in any field isn't standardized across the state whatsoever, so I'm coming from a very different perspective on your point that "people involved in pre-college maths education have defined a set of rules that at best approximate the conventions of maths as actually practised." There's no one U.S. educational system, but even if you're referring to how math is taught in your country's system/main educational system, I'm still not clear on 1) what set of rules you're referring to (e.g. the officially instructed sequence for how to approach a math problem with multiple operations?), 2) what are the conventions of maths as actually practiced, and 3) how the conventions of math as practiced differ from the rules defined in pre-college math education. My reading comprehension is usually not too shabby, but I genuinely have no idea what you're trying to communicate and would be down to see some elaboration.
posted by rather be jorting at 9:47 PM on August 2 [1 favorite]

I'm taking syllabus as being the set of things that are imposed on teachers as "this is what you must teach" usually by either an exam board ('this is our test, and these are the things you must teach for it') or a government body (UK national curriculum, for example) or sometimes, individual schools, school groups, systems, head teachers, etc (though the latter usually just pick from available syllabuses, rather than authoring them from scratch, where they are given this responsibility). No matter where the syllabus comes from though, it had to be written by someone to begin with. That is the "syllabus author" - whether an individual or group.

Teachers may not be directly responsible for developing BODMAS or whatever, and treating it as a rule rather than an approximation of the convention, because they're teaching a syllabus. That's great, but it doesn't actually change the fact that somebody involved in education did make that decision (and it seems to be pretty damn commonplace across many education systems in the world) and the implications are still that "In this case though it seems like pre-college math educators have invented a convention that bears only a loose relationship to how math is actually written by practitioners."

Like, it's not about who exactly is to blame here, at least for me. It's about the outcome. If you're adamant that teachers aren't responsible for it because they have to teach what's in the textbook, great, then it's textbook authors. If they aren't responsible, because they have to put it whatever is in the test/syllabus/curriculum, then the person or organisation who defined the syllabus is. You can quibble over exactly who the blame goes to (and this will vary by school system or even individual school, depending on where in the world you are) but that isn't the point for me. The point is that someone in the business of pre-college education came up with this, and it's hard to chase it back further than whoever defines (or authors) the syllabus/curriculum.
posted by Dysk at 12:51 AM on August 3 [1 favorite]

Dysk, if it's helpful, at least in the part of the US public education system in which I'm situated:

a syllabus refers to a printed document produced by (usually) teachers or (sometimes) departments/schools, mainly listing class policies, and having little to no reflection on content.

a curriculum refers to a sequence of lessons (or materials on which to base lessons), which, in most cases is not of local origin--that is to say it's usually bought from something like a textbook company

an objective is a specific task or skill students should learn within a lesson or unit of lessons

a standard is a broad-scope statement of a set of interrelated skills and competencies a student should obtain in a given grade level; in much of the US, for math and ELA, these are (or are very heavily based on) Common Core, which is not really modifiable at this point since it was such a juggling act to get all the different states to buy in (and many haven't)

The issue with this is that standards are defined by state legislatures, but are so broad they don't, for instance, tell you to teach PEMDAS/BODMAS/GEMDAS/GEMS/pick-your-poison, so it's hard to say that that level invented this idea. Meanwhile, standardized tests have to actually be implemented, so they are based on whatever smaller objectives their authors believe fall under the standards, often coming up with wildly different interpretations (PARCC, MCAS, Regents, just as three examples, look almost nothing alike despite nominally testing the same math standards). Nonetheless, none of their authors are really responsible for deciding "what is taught" since they don't even touch instruction

Another level down, curriculum authors want to make their math make sense and work, but ultimately will be judged by the standardized test scores it brings, so their hands are somewhat tied in what they can actually teach--if it strays too far from the tests, districts/systems won't buy the curriculum. Finally, teachers, when they're even given the autonomy to modify the lessons, are still usually held pretty tightly to following the curriculum they've been given, which doesn't really give them room to change instruction substantially.

The long and short is, at least in the US, no one is in charge of whether or how PEMDAS gets taught. No one person or group has the authority to make that kind of decision, and so there's not really a means by which instruction could change. This has been a repeating theme in US math education--we're still super convinced that the point of all math before college is to pass Calculus 1, which means we focus like crazy on functions and graphs and decontextualized symbolic manipulation, but we've almost entirely stripped out statistics, discrete math, number theory, propositional logic, etc. But no one is in a position to change this, without a pretty substantial restructuring of how education is structured and controlled in the US.
posted by thegears at 5:37 AM on August 3 [7 favorites]

See, that's a pretty different and very much more convoluted system than both the state and private/independent school systems I'm familiar with, where syllabus and curriculum refer to the set of what is to be taught, and are often set by a central authority (government education department, exam board, textbook publisher, or individual school or school group usually choosing from those available from exam boards or textbook publishers) who is directly responsible for the content.

I still think the basic point stands, even if the blame is so diffuse as to be impossible to mete out (beyond saying "the system"). Like I said, I'm not interested in who's to blame, just in the outcome: that conventions of mathematics notation are taught as hard and fast rules of mathematics itself. That much is still true in pretty much every system and curriculum I've had any kind of contact with.
posted by Dysk at 5:55 AM on August 3 [1 favorite]

I don't see that as a problem, personally. In my opinion, it would be very confusing for a large percentage of schoolkids in the USA to be explicitly taught that the mathematics notation conventions they are learning are just conventions and that they can and do and will change. The order of operations imperfectly addresses how adults who use math simplify numeric expressions, but that's okay because it does have value in mathematics education geared towards young children, for reasons I've already mentioned.
posted by 23skidoo at 6:54 AM on August 3

Why not teach the notation that people actually use, which has significantly fewer opportunities for ambiguity, and doesn't require you to memorize a secret decoder mnemonic because the visual layout largely reflects the intended order of operations?

And I realize that changing how things are taught involves fighting against a great deal of institutional inertia, so I understand that as a practical matter nothing is likely to change any time soon, but I'm a bit baffled by people who defend the "elementary-school" notation ("÷×") based on arguments that it's somehow easier to understand, or less likely to confuse children when they see fractions, when in fact it seems entirely the opposite to me: the "college" notation makes it clear that fractions aren't some esoteric other thing, but simply division.
posted by Pyry at 9:03 AM on August 3 [2 favorites]

Without PEMDAS it's not obvious that 3x² isn't (3x)². It seems more consistent to teach PEMDAS rather than later teaching really specific rules for polynomials.

If you evaluate 3x² with x=2, you'll get an arithmetic expression like 3(2)² and now you need to apply PEMDAS or something basically equivalent (PEMAS?) so you don't get 36.

The obelus/division symbol can, of course, die in a fire.
posted by BungaDunga at 9:10 AM on August 3 [1 favorite]

2+3x² is evaluated right to left, x²+3 is evaluated left to right. You could come up with some other rule that explains why it works like that, but PEMDAS seems good enough as long as you never introduce the obelus and always write things with the fraction bar, which slots in as a kind of parenthesis so you evaluate the expressions above and below the bar before you divide.
posted by BungaDunga at 9:16 AM on August 3

That is to say: if the expression 2+3*2² equals 14 because of PEMDAS, then it's much easier to evaluate 2+3x² for some x correctly. You really need the same rules to apply in arithmetic as they do in algebra or things will get hopelessly confusing.
posted by BungaDunga at 9:20 AM on August 3

Why not teach the notation that people actually use, which has significantly fewer opportunities for ambiguity, and doesn't require you to memorize a secret decoder mnemonic because the visual layout largely reflects the intended order of operations?

And I realize that changing how things are taught involves fighting against a great deal of institutional inertia, so I understand that as a practical matter nothing is likely to change any time soon, but I'm a bit baffled by people who defend the "elementary-school" notation ("÷×") based on arguments that it's somehow easier to understand, or less likely to confuse children when they see fractions, when in fact it seems entirely the opposite to me: the "college" notation makes it clear that fractions aren't some esoteric other thing, but simply division.

1) At some point, people do learn those notations. Expressing things using the ÷ symbol isn't really found in textbooks all that much after like 5th or 6th grade (in the USA) except for like probably one question on a standardized test in high school maybe that shows you've learned the order of operations. Like, by middle school grades and up, I'd expect to see the notations you mentioned show up exclusively in textbooks.

2) Okay, so why teach using the ÷ at all? Because it's used overwhelmingly (in the USA) on calculators, and calculators are useful educational tools, and having calculators use the same symbols and notation as the symbols and notation kids see in their math classes can help young kids make sense of math.
posted by 23skidoo at 10:36 AM on August 3 [1 favorite]

Or rather, having math classes use the same notation as found on calculators can help young kids make sense of math.
posted by 23skidoo at 10:51 AM on August 3

Eh. The ÷ symbol has been around way longer than calculators, and there's some research to suggest that calculators are maybe not very pedagogically useful. I can see that it might be easier for kids/students just learning arithmetic to have the same placement format with all arithmetic operations though. If we write 2+3 and 3-1 and 4x5 or 4•5 or whatever, then having the two operands above/below each other for division instead of beside each other might confuse arithmetic learners. Though, personally, I'd like to see both subtraction and division be more obviously distinct from addition and multiplication - in both cases, these two inverse operations don't follow all of the same rules or properties as the in some sense more primary operations they are based on (addition and multiplication). Having the same in-line notation for subtraction and division as we have for addition and multiplication I think contributes a little bit to some common student confusions about subtraction and division.
posted by eviemath at 11:04 AM on August 3

Oh, and I just thought of one more reason why the ÷ is useful for students up till the 5th or 6th grade: like, they teach division of fractions around then, and a fraction that has a fraction in the numerator and a fraction in the denominator can be very intimidating and hard to understand for students at those grade levels- writing an expression as something like (3/4) ÷ (1/8) is way less intimidating than (3/4)/(1/8).
posted by 23skidoo at 11:07 AM on August 3 [3 favorites]

In this case though it seems like pre-college math educators have invented a convention that bears only a loose relationship to how math is actually written by practitioners.

There are a couple differences, which the puzzle we're discussing specifically exploits, but on the whole I'd say that "only a loose relationship" is hyperbole.

And worse, they seem to teach the idea that mathematical notation is universal and fixed, whereas in practice papers invent or use unconventional notation all the time, especially in newer subdomains which have had less time to settle on notation.

This is the more relevant detail. I've had various experiences at various schools where I was a student, but learning math as a primarily rote memorization topic without creativity or play or choices or a history of humans actually developing the ideas and definitions and conventions used or there being answers to the questions "why is such-and-such like that? or "why does it work?" seems to be unfortunately common still. There is use in students learning conventions that facilitate communication about precise mathematical operations or expressions or ideas, and there is utility to having some basic "math facts" from the last step one learned when going on to the next step. But culturally we (North America, and several other places around the world) tend to suffer under a number of myths about math (that it is a fixed body of knowledge, that it requires genius or innate talent, etc.) that tend to turn "this is a convention -- it could just as easily have been different -- but it's useful in practice for us to all use the same convention" into "this is the One True Way That It Must Be Done".
posted by eviemath at 11:25 AM on August 3 [1 favorite]

thegears, thanks for writing out such a clear breakdown!

Dysk, thanks for clearing up what you were referring to with your usage of the word syllabus, which, yeah, ended up being not easily one-to-one converted to a U.S. equivalent, as thegears' comment helped illustrate above. I don't really have much of an opinion regarding your point about notations being taught as rules, but I do hope it's now clearer why your previous equivocation of textbook authors with syllabus authors was so confusing earlier, and how such an equivocation can be perceived as a semantic put-down rather than a simple difference of word choice.

Just speaking for myself, as a casual reader of the thread from the U.S., it came across as weirdly belittling/condescending when you commented, "Substitute in "pre-college maths textbook/syllabus authors" if it helps you understand the point." I saw a false equivalency comparing 1) a profession focused on the substantive content of educational material to 2) a relatively insubstantial schedule-editing task unrelated to the actual substance or decision-making involved with teaching math. Although I figured the word "syllabus" probably meant something different to you than to a person in the U.S., without having a clearer idea for what that different meaning was, it looked like the thread was taking a nasty turn out of seemingly nowhere. So then, my next thoughts turned to, "whoa what's going on? Where is everybody coming from? How can we figure out (in this thread about clarity and lack of clarity in math notation and communication) clearer ways of communicating to each other about whatever it is we're communicating about?" But now I have a better idea where you're coming from re: approaches to math instruction.

Anyway, I myself had a somewhat chaotically experimental pre-college math education until AP Calculus and I don't remember how I was taught multiplication or division or PEMDAS as a kid, so I'm mostly just reading along whenever I see a new update to the thread in my Recent Activity and seeing what other people have to say.

P.S. BungaDunga, I think this thread is also the first time I learned that the division sign is called an obelus, which is leading me down a fascinating wiki read. "Originally this sign (or a plain line) was used in ancient manuscripts to mark passages that were suspected of being corrupted or spurious; the practice of adding such marginal notes became known as obelism. The dagger symbol, also called an obelisk, is derived from the obelus and continues to be used for this purpose."
posted by rather be jorting at 11:31 AM on August 3 [3 favorites]

writing an expression as something like (3/4) ÷ (1/8) is way less intimidating than (3/4)/(1/8).

'Course, then you simply get different common student errors, such as students trying to use commutative properties when they don't apply. And the fact that arithmetic with fractions is a point of struggle for many students might have something to do with having to adjust to new notation at the same time as learning new types of numbers and operations with them? I'd be happier to see both subtraction and division use some notational format that differs from the in-line format for addition and multiplication right from the beginning -- but the same format as each other.

(Background: it is not my area of research specialization - not a math ed researcher in general; but I do teach my department's course in mathematics knowledge for teaching for university students planning on careers in elementary education, and so try to keep somewhat up to date or informed about results from that research area.)
posted by eviemath at 11:33 AM on August 3

answers to the questions "why is such-and-such like that? or "why does it work?" seems to be unfortunately common still.

I never got a decent explanation of why division by zero is undefined until I was about 30, and found one in a Calculus intro from Dover
posted by thelonius at 11:40 AM on August 3 [1 favorite]

rather be jorting, thanks. That was a really nice breakdown of the communication hiccup in the thread and I'm glad you made it really clear.
posted by biogeo at 11:57 AM on August 3 [1 favorite]

Eh. The ÷ symbol has been around way longer than calculators, and there's some research to suggest that calculators are maybe not very pedagogically useful. I can see that it might be easier for kids/students just learning arithmetic to have the same placement format with all arithmetic operations though. If we write 2+3 and 3-1 and 4x5 or 4•5 or whatever, then having the two operands above/below each other for division instead of beside each other might confuse arithmetic learners.

Yeah, that's the thing- regardless of how a certain thing is presented in elementary math classes, there will always be good things and bad things about the way that thing is presented. I think the ÷ symbol has value in elementary math education, but that doesn't mean that I think that all other ways of presenting division are valueless. It's always going to be a balancing act between the needs/wants of students who may struggle with math, students who excel in math classes, educators who know math well, educators who don't know math well, bureaucrats, textbook authors, people who research math education, parents who know math well, and parents who don't know math well (and it's really hard to address the needs/wants all those people at the same time).
posted by 23skidoo at 3:40 PM on August 3 [1 favorite]

In construction there's the notion of falsework, temporary structures that support further work and are removed later. The ÷ and × symbols (for arithmetical division and multiplication) is mathematical falsework that's useful for teaching introductory concepts but is abandoned later (calculators aside). When using the fraction bar, as actual mathematics does, there's no ambiguity. Mixing falsework notations and actual mathematical notations is not something covered under any set of rules because it should never come up, and the original question is deeply flawed. It's fundamentally unfair to pose it to students at any level.
posted by sjswitzer at 2:26 AM on August 5 [3 favorites]

(I make the "arithmetical" stipulation because there can be a distinction between dot (∙) and cross (×) products. Scalar products are represented by juxtaposition. Although I've never seen a "rule," it's well understood that all products have precedence over sums.)
posted by sjswitzer at 2:44 AM on August 5

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