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).
It isn't, because the 'currently taught rules' are on a case-by-case basis and each teacher defines this area themselves. Strong juxtaposition isn't already taught, and neither is weak juxtaposition. That's the whole point of the argument.
See this part of my comment: "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)."
You're claiming the post is wrong and saying it doesn't have any textbook citation (which is erroneous in and of itself because textbooks are not the only valid source) but you yourself don't put down a citation for your own claim so... citation needed.
In addition, this issue isn't a mathematical one, but a grammatical one. It's about how we write math, not how math is (and thus the rules you're referring to such as the Distributive Law don't apply, as they are mathematical rules and remain constant regardless of how we write math).
Nope. Teachers can decide how they teach. They cannot decide what they teach. The have to teach whatever is in the curriculum for their region.
That's because neither of those is a rule of Maths. The Distributive Law and Terms are, and they are already taught (they are both forms of what you call "strong juxtaposition", but note that they are 2 different rules, so you can't cover them both with a single rule like "strong juxtaposition". That's where the people who say "implicit multiplication" are going astray - trying to cover 2 rules with one).
Yep, saw it, and weak juxtaposition would break the existing rules of Maths, such as The Distributive Law and Terms. (Re)learn the existing rules, that is the point of the argument.
Well that part's easy - I guess you missed the other links I posted. Order of operations thread index Text book references, proofs, the works.
Maths isn't a language. It's a group of notation and rules. It has syntax, not grammar. The equation in question has used all the correct notation, and so when solving it you have to follow all the relevant rules.
Yes, teachers have certain things they need to teach. That doesn't prohibit them from teaching additional material.
You argue about sources and then cite yourself as a source with a single reference that isn't you buried in the thread on the Distributive Law? That single reference doesn't even really touch the topic. Your only evidence in the entire thread relevant to the discussion is self-sourced. Citation still needed.
You can argue semantics all you like. I would put forth that since you want sources so much, according to Merriam-Webster, grammar's definitions include "the principles or rules of an art, science, or technique", of which I think the syntax of mathematics qualifies, as it is a set of rules and mathematics is a science.
Correct, but it can't be something which would contradict what they do have to teach, which is what "weak juxtaposition" would do.
I see you didn't read the whole thread then. Keep going if you want more. Literally every Year 7-8 Maths textbook says the same thing. I've quoted multiple textbooks (and haven't even covered all the ones I own).
Actually you'll find that assertion is hotly debated.
Citation needed.
If I have to search your 'source' for the actual source you're trying to reference, it's a very poor source. This is the thread I searched. Your comments only reference 'math textbooks', not anything specific, outside of this link which you reference twice in separate comments but again, it's not evidence for your side, or against it, or even relevant. It gets real close to almost talking about what we want, but it never gets there.
But fine, you reference 'multiple textbooks' so after a bit of searching I find the only other reference you've made. In the very same comment you yourself state "he says that Stokes PROPOSED that /b+c be interpreted as /(b+c). He says nothing further about it, however it's certainly not the way we interpret it now", which is kind of what we want. We're talking about x/y(b+c) and whether that should be x/(yb+yc) or x/y * 1/(b+c). However, there's just one little issue. Your last part of that statement is entirely self-supported, meaning you have an uncited refutation of the side you're arguing against, which funnily enough you did cite.
Now, maybe that latter textbook citation I found has some supporting evidence for yourself somewhere, but an additional point is that when providing evidence and a source to support your argument you should probably make it easy to find the evidence you speak of. I'm certainly not going to spend a great amount of effort trying to disprove myself over an anonymous internet argument, and I believe I've already done my due diligence.
So you think it's ok to teach contradictory stuff to them in Maths? 🤣 Ok sure, fine, go ahead and find me a Maths textbook which has "weak juxtaposition" in it. I'll wait.
So you're telling me you can't see the Maths textbook screenshots/photo's?
Lennes was complaining that literally no textbooks he mentioned were following "weak juxtaposition", and you think that's not relevant to establishing that no textbooks used "weak juxtaposition" 100 years ago?
It's in literally the first textbook screenshot, which if I'm understanding you right you can't see? (see screenshot of the screenshot above)
Ah, no. Lennes was complaining about textbooks who were obeying Terms/The Distributive Law. His own letter shows us that they all (the ones he mentioned) were doing the same thing then that we do now. Plus my first (and later) screenshot(s).
Also it's in Cajori, but I didn't find it until later. I don't remember what page it was, but it's in Cajori and you have the reference for it there already.
Well I'm not sure how you didn't see all the screenshots. They're hard to miss on my computer!
You haven't provided a textbook that has strong juxtaposition.
That's not a source, that's a screenshot. You can't look up the screenshot, you can't identify authors, you can't check for bias. At best I can search the title of the file you're in that you also happened to screenshot and hope that I find the right text. The fact that you think this is somehow sufficient makes me question your claims of an academic background, but that's neither here nor there. What does matter is that I shouldn't have to go treasure hunting for your sources.
And, to blatantly examine the photo, this specific text appears to be signifying brackets as their own syntactic item with differing rules. However, I want to note that the whole issue is that people don't agree so you will find cases on both sides, textbook or no.
You are welcome to cite the specific wording he uses to state this. As far as I can tell, at least in the excerpt linked, there is no such complaint.
Here you go - I found I did save a screenshot of Cajori saying ab and (ab) are the same thing - I didn't think I had.
I told you, in my thread - multiple ones. You haven't provided any textbooks at all that have "weak juxtaposition". i.e. you keep asking me for more evidence whilst never producing any of your own.
I didn't "just happen" to include the name of the textbook and page number - that was quite deliberate. Not sure why you don't want to believe a screenshot, especially since you can't quote any that have "weak juxtaposition" in the first place.
BTW I just tried Googling it and it was the first hit. You're welcome.
You don't - the screenshots of the relevant pages are right there. You're the one choosing not to believe what is there in black and white, in multiple textbooks.
Yeah, I wrote about inconsistency in textbooks here (also includes another textbook saying you have to expand brackets first), but also elsewhere in the thread is an example where they have been consistent throughout. Regardless of when they remove brackets, in every single case they multiply the coefficient over what's inside the brackets as the first step (as per BEDMAS, and as per the screenshot in question which literally says you must do it before you remove brackets).
People who aren't high school Maths teachers (the ones who actually teach this topic). Did you notice that neither The Distributive Law nor Terms are mentioned at any point whatsoever? That's like saying "I don't remember what I did at Xmas, so therefore it's ambiguous whether Xmas ever happened at all, and anyone who says it definitely did is wrong".
So what do you think he is complaining about?
You seem to have missed the point. I'm holding you to your own standard, as you are the one that used evidence as an excuse for dismissal first without providing evidence for your own position.
You seem to have missed the point. You're providing a bad source and expecting the person you're arguing against to do legwork. I never said I couldn't find the source. I'm saying I shouldn't have to go looking.
You've provided a single textbook, first of all. Second of all, the argument is that both sides are valid and accepted depending on who you ask, even amongst educated echelons. The fact there exists textbooks that support strong juxtaposition does nothing to that argument.
But you want some evidence, so here's an article from someone who writes textbooks speaking on the ambiguity. Again, the ambiguity exists and your claim that it doesn't according to educated professors is unsubstantiated. There are of course professors who support strong juxtaposition, but there are also professors who support weak juxtaposition and professors that merely acknowledge the ambiguity exist. The rules of mathematics you claim are set in stone aren't relevant (and aren't as set in stone as you imagine) but that's not entirely relevant. What is relevant is there is an argument and it's not just uneducated folk mistaking the 'truth'.
You are correct, I suppose a mathematics professor from Harvard (see my previous link for the relevant discussion of the ambiguity) isn't at the high school level.
But wait, there's more. Here's another source from another mathematics professor. This one 'supports' weak juxtaposition but really mostly just points at the ambiguity. Which again, is what I'm going for, that the ambiguity exists and one side is not immediately justified/'correct'.
That's a leading question and is completely unhelpful to the discussion. I asked you to point out where exactly, and with what wording, your position is supported in the provided text. Please do that.
You know full well it's all in my thread. Where's yours?
You didn't have to go looking - you could've just accepted it at face-value like other people do.
No, multiple textbooks. If you haven't seen the others yet then keep reading. On the other hand you haven't provided any textbooks.
But they're not. The other side is contradicting the rules of Maths. In a Maths test it would be marked as wrong. You can't go into a Maths test and write "this is ambiguous" as an answer to a question.
Not high school textbooks! Talk about appeal to authority.
Yep, seen it before. Note that he starts out with "It is not clear what the textbook had intended with the 3y". How on Earth can he not know what that means? If he just picked up any old high school Maths textbook, or read Cajori, or read Lennes' letter, or even just asked a high school teacher(!), he would find that every single Maths textbook means exactly the same thing - ab=(axb). Instead he decided to write a long blog saying "I don't know what this means - it must be ambiguous".
Not only that, but he also didn't know how to handle x/x/x, which shows he doesn't remember left associativity either. BTW it's equal to x/x² (which is equal to 1/x).
...amongst people who have forgotten the rules of Maths. The Maths itself is never ambiguous (which is the claim many of them are making - that the Maths expression itself is ambiguous. In fact the article under discussion here makes that exact claim - that it's written in an ambiguous way. No it isn't! It's written in the standard mathematical way, as per what is taught from textbooks). It's like saying "I've forgotten the combination to my safe, and I've been unable to work it out, therefore the combination must be ambiguous".
Thank you. I just commented to someone else last night, who had noticed the same thing, I am so tired of people quoting University people - this topic is NOT TAUGHT at university! It's taught by high school teachers (I've taught this topic many times - I'm tutoring a student in it right now). Paradoxically, the first Youtube I saw to get it correct (in fact still the only one I've seen get it correct) was by a gamer! 😂 He took the algebra approach. i.e. rewrite this as 6/2a where a=1+2 (which I've also used before too. In fact I did an algebraic proof of it).
The side which obeys the rules of Maths is correct and the side which disobeys the rules of Maths is incorrect. That's why the rules of Maths exist in the first place - only 1 answer can be correct ("ambiguity" people also keep claiming "both answers are correct". Nope, one is correct and one is wrong).
Twice I said things about it and you said you didn't believe my interpretation is correct, so I asked you what you think he's saying. I'm not going to go round in circles with you just disagreeing with everything I say about it - just say what YOU think he says.
I could also walk off a cliff, doesn't mean I should. Sources are important not just for what they say but how they say it, where they say it, and why they say it.
Yes, that is your claim which you have yet to prove. You keep reiterating your point as if it is established fact, but you haven't established it. That's the whole argument.
Literally just give me a direct quote. If you're using it as supporting evidence, tell me how it supports you. If you can't even do that, it's not supporting evidence. I don't know why you want me to analyze it, you're the one who presented it as evidence. My analysis is irrelevant.
I was being sarcastic. If you truly think highschool teachers who require almost no training in comparison to a Phd are more qualified... I have no interest in continuing this discussion. That's simply absurd, professors study every part of mathematics (in aggregate), including the 'highschool' math, and are far more qualified than any highschool teacher who is not a Phd. This is true of any discipline taught in highschool, a physics professor is much better at understanding and detailing the minutiae of physics than a highschool physics teacher. To say a teacher knows more than someone who has literally spent years of their life studying and expanding the field when all the teacher has to do is teach the same (or similar) curriculum each and every year is... insane--especially when you've been holding up math textbooks as the ultimate solution and so, so many of them are written by professors.
I want to point out that your only two sources, both a screenshot of a textbook, (yes, those are your only sources. You've given 4, but one I've repeatedly asked about and you've refused to point out a direct quote that provides support for your argument, another I dismissed earlier and I assume you accepted that seeing as you did not respond to that point) does not state the reasoning behind its conclusion. To me that's far worse than a professor who at least says why they've done something.
I've given 3 sources, all of which you dismiss simply because they're not highschool textbooks... y'know, textbooks notorious for over-simplifying things and not giving the logic behind the answer. I could probably find some highschool textbooks that support weak juxtaposition if I searched, but again that's a waste of money and time. You don't seem keen on acknowledging any sort of ambiguity here and constantly state it goes against the rules of math, without ever providing a source that explains these rules and how they work so as to prove only strong juxtaposition makes sense/works. If you're really so confident in strong juxtaposition being the only way mathematically, I expect you to have a mathematical proof for why weak juxtaposition would never work, one that has no flaws. Otherwise, at best you have a hypothesis.
None of which you've addressed since I gave you the source. Remember when you said this...
So, did you do that once I gave you the link? And/or are you maybe going to address "what they say but how they say it, where they say it, and why they say it" in regards to the link I gave you?
What they teach in Maths textbooks aren't facts? Do go on. 😂
I did, and you've apparently refused to read the relevant part.
You know not all university lecturers do a Ph.D. right? In which case they haven't done any more study at all. But I know you really wanna hang on to this "appeal to authority" argument, since it's all you've got.
Yeah I saw that coming once I gave you the link to the textbook.
...when they were in high school.
There you go. Welcome to why high school teachers are the expert in this field.
So wait, NOW you're saying textbooks ARE valid in what they say? 😂
All that points out is that you didn't even read THIS thread properly, never mind the other one. Which two are they BTW? And I'll point out which ones you've missed.
Well, I'll use your own logic then to take that as a concession, given how many of my points you didn't respond to (like the textbook that I gave you the link to, and the Cajori ab=(ab) one, etc.).
3 articles you mean.
...all of them have forgotten about The Distributive Law and Terms., which make the expression totally unambiguous. Perhaps you'd like to find an article that DOES talk about those and ALSO asserts that the expression is "ambiguous"? 😂 Spoiler alert: every article, as soon as I see the word "ambiguous" I search the text for "distributive" and "expand" and "terms" - can you guess what I find? 😂 Hint: Venn diagram with little or no overlap.
Do you wanna bet on that? 😂
They're in my thread, if you'd bothered to read any further. By your own standards, 😂I'll take it that you concede all of my points that you haven't responded to.
You know some things are true by definition, right, and therefore don't have a proof? 1+1=2 is the classic example. Or do you challenge that too?
So do YOU have a hypothesis then? How "weak juxtaposition" could EVER work given "strong juxtaposition" is the only type ever used in any of the rules of Maths? I'll wait for your proof...
At this point you're just ignoring whatever I say and I see no point in continuing this discussion. You haven't responded to what I've said, you've just stated I'm wrong and to trust you on that because somewhere prior you said so. Good luck with convincing anyone that way.
You know EXACTLY where I said those things, and you've been avoiding addressing them ever since because you know they prove the point that #MathsIsNeverAmbiguous See ya.
Also noted that you've declined on taking on that bet I offered.
P.S. if you DID want to indicate "weak juxtaposition", then you just put a multiplication symbol, and then yes it would be done as "M" in BEDMAS, because it's no longer the coefficient of a bracketed term (to be solved as part of "B"), but a separate term.
6/2(1+2)=6/(2+4)=6/6=1
6/2x(1+2)=6/2x3=3x3=9
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.
You think a Maths teacher doesn't know the difference?
Yes there is - the underlying Maths. 2x3 is short for 2+2+2, which is therefore why you have to expand multiplications before doing additions. If you "chose" to interpret 2+3x4 (which we KNOW is equal to 14, because 3x4=3+3+3+3 by definition) as (2+3)x4, you would get 20, which is clearly wrong, since 20 isn't equal to 14.
No that's right, because it IS written differently in different languages, but regardless of how you write it, it doesn't change that 2+3x4=14 - the underlying Maths doesn't change regardless of how you decide to write it. Maths is literally universal.
It's not a choice, it's a consequence of the fact that x is shorthand for +. i.e. 2x3=2+2+2.
It is a consequence of the definitions of what each operator does. If x is a contraction of +, then we have to expand x before we do +. If it were the other way around then we'd have to do it the other way around. Anything which is a contraction of something else has to be expanded first.
Hey man, if you want to resort to some weird appeal to authority argument despite having clear examples against what you are saying - go for it. You can choose to die on that hill if you want to.
Which are where, exactly? You haven't presented any. You haven't, for example, shown how one can make (2+3)x4=14.
re: appeal to authority
The examples I gave were that the expansion of brackets would be done differently if the order of operations was "PESADM"; and I also drew your attention to the fact that reverse polish notation exists, in which there are no brackets at all and the order of operation is entirely determined by the order that operators appear, with no hierarchy of operations. As for your appeal to authority, let me just say that your level of qualification on this topic is not above mine. It adds no weight whatsoever to your argument.
I just glanced at your post history to get a sense of why you were so engaged in this. I was a bit startled to see that you've been on a bit of a posting spree in this thread, which I point out to you is a 3 month old post on a 'memes' channel. I see you've taken issue with a lot of what people have said here. My suggestion to you now is that there probably won't be a lot of engagement in this thread from this point on. So perhaps you should just ponder what is said, and prepare yourself again for next time this comes up. Perhaps you can start by seeing if you can get a consensus amongst fellow experts in a maths channel or something, because at the moment it seems like you're on your own.
Yep I read it, and no it wouldn't. Expanding Brackets - or in the case of this mnemonic Parentheses - is done as part of B/D (as the case may be). i.e. expanding brackets isn't "multiplication" (no multiplication sign), but solving brackets (there are brackets there), which always come first in all the mnemonics.
...but is not taught in high school.
Maybe not, but it means it's not an "appeal to authority" (as per screenshot). Maths teachers ARE an authority on Maths. The most common appeal to authority I see from people is claiming that someone (not them) is a University professor, and "they would know". No, they wouldn't - this topic isn't taught at university - it's taught in high school.
I'm a teacher. You say you're on the same level as me - don't you like to teach people what's correct?
Which will show up in search results for all eternity (it's how I found it - I was looking for something else!).
Got another 12 responses after yours. But the point is I'm not even LOOKING for responses, just to correct misinformation. As a teacher (a Maths teacher?) have you not had people say to you "But Google says"? I certainly have. It's the bane of my professions.
Did you read my thread? Maths textbooks, calculators, proofs, etc. Also, someone else said what you just did, asked a Maths teacher, and was told I was correct, then was man enough to go back and edit his posts and admit I was correct and specifically said "SmartmanApps is not on his own with this".
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.
Ok, that's a start. In your simple example they are all equal, but they aren't all the same.
yn+y - 2 terms
y(n+1) - 1 term
y×(n +1) - 2 terms
To see the difference, now precede it with a division, like in the original question...
1÷yn+y=(1/yn)+y
1÷y(n+1)=1/(yn+y)
1÷y×(n +1)=(n +1)/y
Note that in the last one, compared to the second one, the (n+1) is now in the numerator instead of in the denominator. Welcome to why having the (2+2) in the numerator gives the wrong answer.
Good example wish we had better math format.
The granger issue is I thought multiple always happens first. But apparently it's what's left side first.
Multiplication and division are equal precedence (and done left to right) if that's what you're talking about, but the issue is that a(b+c) isn't "multiplication" at all, it's a bracketed term with a coefficient which is therefore subject to The Distributive Law, and is solved as part of solving Brackets, which is always first. Multiplication refers literally to multiplication signs, of which there are none in the original question. A Term is a product, which is the result of a multiplication, not something which is to be multiplied.
If a=2 and b=3, then...
axb=2x3 - 2 terms
ab=6 - 1 term
That is what I am talking about. I would have got 1 by doing 2(2+2) = 8 first. Not because of bracket but because of "implied multiplication."
What I am learning here: 8÷2(2+2) is not same as 8÷2×(2+2)
For several reasons:
Yeah, right answer but wrong reason. There's no such thing as implicit multiplication.
Correct, and that's because of Terms - 8÷2(2+2) is 2 terms, with the (2+2) in the denominator, but 8÷2×(2+2) is 3 terms, with the (2+2) in the numerator... hence why people get the wrong answer when they add an extra multiply in.
Right, because it's not "multiplication" at all (only applies literally to multiplication signs), it's a coefficient of a bracketed term, which means we have to apply The Distributive Law as part of solving Brackets.
Yeah, the actual rule is Left associativity, and going left to right is the easy way to obey that.
Lol only tool 30 years to get thos far on basics.
Just proves, never too late to learn :-)
Thanks for the effort!
You're welcome.