I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It’s about a 30min read so thank you in advance if you really take the time to read it, but I think it’s worth it if you joined such discussions in the past, but I’m probably biased because I wrote it :)
Having read your article, I contend it should be:
P(arentheses)
E(xponents)
M(ultiplication)D(ivision)
A(ddition)S(ubtraction)
and strong juxtaposition should be thrown out the window.
Why? Well, to be clear, I would prefer one of them die so we can get past this argument that pops up every few years so weak or strong doesn’t matter much to me, and I think weak juxtaposition is more easily taught and more easily supported by PEMDAS. I’m not saying it receives direct support, but rather the lack of instruction has us fall back on what we know as an overarching rule (multiplication and division are equal). Strong juxtaposition has an additional ruling to PEMDAS that specifies this specific case, whereas weak juxtaposition doesn’t need an additional ruling (and I would argue anyone who says otherwise isn’t logically extrapolating from the PEMDAS ruleset). I don’t think the sides are as equal as people pose.
To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).
But again, I really don’t care. Just let one die. Kill it, if you have to.
It’s like using literally to add emphasis to something that you are saying figuratively. It’s not objectively “wrong” to do it, but the practice is adding uncertainty where there didn’t need to be any, and thus slightly diminishes our ability to communicate clearly.
I think anything after (whichever grade your country introduces fractions in) should exclusively use fractions or multiplication with fractions to express division in order to disambiguate. A division symbol should never be used after fractions are introduced.
This way, it doesn’t really matter which juxtaposition you prefer, because it will never be ambiguous.
Anything before (whichever grade introduces fractions) should simply overuse brackets.
This comment was written in a couple of seconds, so if I missed something obvious, feel free to obliterate me.
But a fraction is a single term, 2 numbers separated by a division is 2 terms. Terms are separated by operators and joined by grouping symbols.
Except it breaks the rules which already are taught.
But they’re not rules - it’s a mnemonic to help you remember the actual order of operations rules.
Juxtaposition - in either case - isn’t a rule to begin with (the 2 appropriate rules here are The Distributive Law and Terms), yet it refuses to die because of incorrect posts like this one (which fails to quote any Maths textbooks at all, which is because it’s not in any textbooks, which is because it’s wrong).
Division comes before Multiplication, doesn’t it? I know BODMAS.
That makes no sense. Division is just multiplication by an inverse. There’s no reason for one to come before another.
This actually explains alot. Murica is Pemdas but Canadian used Bodmas so multiply is first in America.
As far as I understand it, they’re given equal weight in the order of operations, it’s just whichever you hit first left to right.
Ah, but if you use the rules BODMSA (or PEDMSA) then you can follow the letter order strictly, ignoring the equal precedence left-to-right rule, and you still get the correct answer. Therefore clearly we should start teaching BODMSA in primary schools. Or perhaps BFEDMSA. (Brackets, named Functions, Exponentiation, Division, Multiplication, Subtraction, Addition). I’m sure that would remove all confusion and stop all arguments. … Or perhaps we need another letter to clarify whether implicit multiplication with a coefficient and no symbol is different to explicit multiplication… BFEIDMSA or BFEDIMSA. Shall we vote on it?
Don’t need any extra letters - just need people to remember the rules around expanding brackets in the first place.
Obviously more letters would make the mnemonic worse, not better. I was making a joke.
As for the brackets ‘the rules around expanding brackets’ are only meaningful in the assumed context of our order of operations. For example, if we instead all agreed that addition should be before multiplication, then a×(b+c) would “expand” to a×b+c, because the addition is before multiplication anyway and the brackets do nothing.
Fair enough, but my point still stands.
…then you would STILL have to do multiplication first. You can’t change Maths by simply agreeing to change it - that’s like saying if we all agree that the Earth is flat then the Earth is flat. Similarly we can’t agree that 1+1=3 now. Maths is used to model the real world - you can’t “agree” to change physics. You can’t add 1 thing to 1 other thing and have 3 things now, no matter how much you might want to “agree” that there is 3, there’s only 2 things. Multiplying is a binary operation, and addition is unary, and you have to do binary operators before unary operators - that is a fact that no amount of “agreeing” can change. 2x3 is actually a contracted form of 2+2+2, which is why it has to be done before addition - you’re in fact exposing the hidden additions before you do the additions.
The brackets, by definition, say what to do first. Regardless of any other order of operations rules, you always do brackets first - that is in fact their sole job. They indicate any exceptions to the rules that would apply otherwise. They perform no other function. If you’re going to no longer do brackets first then you would simply not use them at all anymore. And in fact we don’t - when there are redundant brackets, like in (2)(1+2), we simply leave them out, leaving 2(1+2).
I believe you’re conflating the rules of maths with the notation we use to represent mathematical concepts. We can choose whatever notation we like to mean anything we like. There is absolutely nothing stopping us from choosing to interpret a+b×c as (a+b)×c rather than a+(b×c). We don’t even have to write it like that at all. We could write a,b,c×+. (And sometimes people do write it like that.) Notation is just a way to communicate. It represents the maths, but it is not itself the maths. Some notation is more convenient or more intuitive than others. × before + is a very convenient choice, because it easier to express mathematical truths clearly and concisely - but nevertheless, it is still just a choice.
Yeah 100% was not taught that. Follow the pemdas or fail the test. Division is after Multiply in pemdas.
I put the equation into excel and get 9 which only makes sense in bodmas.
It doesn’t make sense in BODMAS either. Expanding Brackets has precedence of… Brackets, not “multiplication” - “Multiplication” refers literally to multiplication signs, of which there are none in this question.
The y(n+1) is same as yn + y if you removed the “6÷” part. It’s implied multiplication.
No, it’s the same as (yn+y). You can’t remove brackets unless there is only 1 term left inside.
…The Distributive Law.
Well I’m not seeing the difference here. Yn+y= yn+y = y(n+1) = y × (n +1) I think we agree with that.