6÷2(1+2)
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 :)
802
Comments488
I beg to differ. You utter fool! You created a comment section yourself on lemmy and you are clearly wrong about everything!
You take the mean of 1 and 9 which is 4.5!
/j
🤣 I wasn't even sure if I should post it on lemmy. I mainly wrote it so I can post it under other peoples posts that actually are intended to artificially create drama to hopefully show enough people what the actual problems are with those puzzles.
But I probably am a fool and this is not going anywhere because most people won't read a 30min article about those math problems :-)
Actually the correct answer is clearly 0.2609 if you follow the order of operations correctly:
6/2(1+2)
= 6/23
= 0.26
Nah man, distribute the 2.
6/2(1+2)
= 6/2+4
= 3+4
= 7
This is like 4st grayed maff.
🤣 I'm not sure if you read the post but I also wrote about that (the paragraph right before "What about the real world?")
I did read the post (well done btw), but I guess I must have missed that. And here I thought I was a comedic genius
@relevants you truly are the smartest of all men
I did (skimmed it, at least) and I liked it. 🙃
Right, because 5 rounds down to 4.5
@Prunebutt meant 4.5! and not 4.5. Because it's not an integer we have to use the gamma function, the extension of the factorial function to get the actual mean between 1 and 9 => 4.5! = 52.3428 which looks about right 🤣
Not sure if sarcastic and woosh, or adding to the joke ಠ_ಠ
The mean of 1 and 9 is 5
woosh
I think you got hit hard by Poe's Law here. Except it's more like people couldn't tell if you were jokingly or genuinely getting your math wrong... Even after you explained you were joking lol
I thought the "/j" tone-tag was enough ;_;
jarcasm?
If one doesn't realize you're op, the entire thing can be interpreted very differently.
Then "Not sure if sarcastic and woosh, or adding to the joke ಠ_ಠ" could be interpreted as something like "I'm not sure if you are adding to the joke and I'm not understanding it".
Stop it Patrick, you're scaring them!
…Because 4 rounds up to 4.5
The answer realistically is determined by where you place implicit multiplication (or "multiplication by juxtaposition") in the order of operations.
Some place it above explicit multiplication and division, meaning it gets done before the division giving you an answer of 1
But if you place it as equal to it's explicit counterparts, then you'd sweep left to right giving you an answer of 9
Since those are both valid interpretations of the order of operations dependent on what field you're in, you're always going to end up with disagreements on questions like these...
But in reality nobody would write an equation like this, and even if they did, there would usually be some kind of context (I.e. units) to guide you as to what the answer should be.
Edit: Just skimmed that article, and it looks like I did remember the last explanation I heard about these correctly. Yay me!
Exactly. With the blog post I try to reach people who already heared that some people say it's ambiguous but either down understand how, or don't believe it. I'm not sure if that will work out because people who "already know the only correct answer" probably won't read a 30min blog post.
Unfortunately these types of viral problems are designed the attract people who think they "know it all", so convincing them that their chosen answer isn't as right as they think it is will always be an uphill challenge
yeah, our math profs taught if the 2( is to be separated from that bracket for the implied multiplication then you do that math first, because the 2(1+2) is the same as (1+2)+(1+2) and not related to the first 6.
So you were taught strong juxtaposition then, where the implicit multiplication takes priority?
if it was 6÷2x(2+1) they suggested do division and mult from left to right, but 6÷2(2+1) implied a relationship between the number outside the parenthesis and inside them, and as soon as you broke those () you had to do the multiplication immediately that is connected to them. Like some models of calculatora do. wasn't till a few yeara ago that I heard people were doing it differently.
Correct! Terms are separated by operators and joined by grouping symbols, so 6÷2x(2+1) is 3 terms - 6, 2, and (2+1) - whereas 6÷2(2+1) is 2 terms - 6 and 2(2+1), and the latter term has a precedence of "brackets", NOT "multiplication". Multiplication refers literally to multiplication signs, which are only present in your first example (hence evaluated with a different order than your second example).
Also noted that the OP has ignored your comment, seeing as how you pointed out the unambiguous way to do it.
You nearly had it. 2(1+2) is the same as (2x1+2x2). The Distributive Law - it's the reverse process to factorising.
Yeah, that's why fractions are good thing.
There's no such thing as "implicit multiplication"
Which is correct, seeing as how we're solving brackets, and brackets always come first.
Which is wrong.
No, they're not. Treating brackets as, you know, brackets, is the only valid interpretation. "Multiplication" refers literally to multiplication signs, of which there are none in this problem.
Yes they would. a(b+c) is the standard way to write a factorised term.
Ackshually, the answer is 4
6÷2*(1+2)
6÷(1+2)*2
6÷(3)*2
2*2
4
You're welcome
psychopath
Psychomath
c/TheyDidAMath
If there are rules about which dot comes first then you are not allowed to do this.
You aren't allowed to do this because division isn't transitive.
In fact you're not allowed to add the multiplication - it breaks up the factorised term, hence gives a different answer.
The rule is you're not allowed to add dots (multiplication) - broke up the factorised term, which is why a different answer.
Typo in article:
Except should be 'accept'.
Not trying to be annoying, but I know people will often find that as a reason to disregard academic arguments.
Thank you very much 🫶. No it's not annoying at all. I'm very grateful not only for the fact that you read the post but also that you took the time to point out issues.
I just fixed it, should be live in a few minutes.
A person not knowing the difference in usage between except and accept sounds like a perfectly reasonable reason to disregard their math skills.
Especially when said person keeps making incorrect statements about Maths and ignores completely what is taught in high school.
The "academic arguments" can be ignored since this is actually high school Maths - it's taught in Year 7-8.
Great write up! The answer is use parentheses or fractions and stop wasting everyone’s time 😅
That's actually a great way of putting it 🤣
Funny how using parentheses gets you the same answer as if implicit multiplication doesn't have a higher order... It's almost like considering implicit multiplication as having an advanced order is an invalid assumption to make when looking at a maths equation.
Edit: I'm wrong, read below.
It's not invalid or even uncommon. It's just not necessarily correct. Implicit multiplication can be used intentionally to differentiate from explicit multiplication and context can suggest there is a difference in priority. For example, a/bc is likely to be read as a/(bc) because the alternative could be written less ambiguously as ac/b. If I wanted to convey to you that multiplication is associative, I might say ab*c = a*bc, and you'd probably infer that I'm communicating something about the order of operations. But relying on context like this is bad practice, so we always prefer to use parentheses to make it explicit.
It's only ambiguous if you don't read left to right. That's a literacy issue not a mathematics one.
It's definitely not a mathematics issue. This all concerns only notation, not math. But it's not a literacy issue either. It's ambiguous in that the concept of a correct order of operations itself is wrong.
Notation is read left to right, reading it in any other order is automatically incorrect. Just like if you read a sentence out of order you won't get it's intention. Like I said, if you actually follow the rules it's almost like implicit multipication having a higher order doesn't work, which makes it illigitimate mathematics.
It's not left to right. a+b*c is unambiguously equal to a+(b*c) and not (a+b)*c.
Correct! "implicit multiplication" is NOT a rule of Maths. It's something made up by those who don't remember all the actual rules, which includes The Distributive Law and Terms.
No it isn't dotnet.social/@SmartmanApps/110819283738912144
What if the real answer is the friends we made along the way?
This is Facebook we are talking about, what friends? Everyone hates everyone on Facebook
That'd be good, but what I've found so far here is a whole bunch of people who don't like being told the actual facts of the matter! 😂
I tried explaining this to people on facebook in 2010 or so.
"You must be fun at parties!"
Bitch, i dont want to attend your lame ass party where people think they know how math works.
I love that the calculators showing different answers are both from the same manufacturer XD
The real lesson here is that clear, unambiguous communication is key.
And just what do you mean by that?
/s
My TI-84 Plus is my holy oracle, I will go with whatever it says.
And then get distracted and play some Doom.
It will give 9, just like my 89 emulator. It treats division like a fraction. For a TI, the entire denominator of a fraction needs to be in parentheses or you get into trouble.
Which is why it gives the wrong answer.
Also you shouldn't be adding a dot between the 2 and the brackets - that also changes the answer.
❤️
TI calcs give the wrong answer, and it's in their manual why - they only follow the Primary School rule ("inside the brackets"), not the High School rule which supersedes it (The Distributive Law).
What's especially wild to me is that even the position of "it's ambiguous" gets almost as much pushback as trying to argue that one of them is universally correct.
Last time this came up it was my position that it was ambiguous and needed clarification and had someone accuse me of taking a prescriptive stance and imposing rules contrary to how things were actually being done. How asking a person what they mean or seeking clarification could possibly be prescriptive is beyond me.
Bonus points, the guy telling me I was being prescriptive was arguing vehemently that implicit multiplication having precedence was correct and to do otherwise was wrong, full stop.
👍 That was actually one of the reasons why I wrote this blog post. I wanted to compile a list of points that show as clear as humanity possible that there is no consensus here, even amongst experts.
That probably won't convince everybody but if that won't probably nothing will.
And I wrote a bunch of fact checks pointing out there is consensus amongst the actual experts - high school Maths teachers and textbook authors, the 2 groups who you completely ignored in your blog post.
When I went to college, I was given a reverse Polish notation calculator. I think there is some (albeit small) advantage of becoming fluent in both PEMDAS and RPN to see the arbitrariness. This kind of arguement is like trying to argue linguistics in a single language.
Btw, I'm not claiming that RPN has any bearing on the meme at hand. Just that there are different standards.
This comment is left by the HP50g crew.
It would be better if we just taught math with prefix or postfix notation, as it removes the ambiguity.
Ambiguity is fine. It would tedious to the point of distraction to enforce writing math without ambiguity. You make note of conventions and you are meant to realize that is just a convention. I'm amazed at the people who are planting their feet to fight for something that what they were taught in third grade as if the world stopped there.
You're right though. We should definitely teach different conventions. But then what would facebook do for engagement?
It already is written without ambiguity.
This is actually taught in Year 7 - the people who only remember the 3rd Grade version of the rules are the ones getting it wrong.
There isn't ambiguity to begin with - just people who have forgotten the rules of Maths.
That's because following the rules of Maths is universally correct.
He was using the wrong words, but he was correct - the actual rules are The Distributive Law and Terms ("implicit multiplication" is a rule made up by those who have forgotten these 2 rules).
Without any additional parentheses, the division sign is assumed to separate numerators and denominators within a complete expression, in which case you would reduce each separately. It's very, very marginally ambiguous at best.
Assumed by whom? Clearly not everyone.
It's what is actually taught in high school, so there are those who remember and those who don't.
You are correct with your definition - Terms are separated by operators and joined by grouping symbols - and it's consequently not ambiguous at all (using so-called "weak juxtaposition" breaks that rule).
Just write it better.
6/(2(1+2))
Or
(6/2)(1+2)
That's how it works in the real world when you're using real numbers to calculate actual things anyways.
It's hilarious seeing all the genius commenters who didn't read the linked article and are repeating all the exact answers and arguments that the article rebuts :)
I'm still not used to having combined image and text posts so I usually don't notice the text portion if it isn't a big ol' wall and I hope I'm not the only one.
❤️ True, but I think one of the biggest problems is that it's pretty long and because you can't really sense how good/bad/convining the text is it's always a gamble for everybody if it's worth reading something for 30min just to find out that the content is garbage.
I hope I did a decent job in explaining the issue(s) but I'm definitely not mad if someone decides that they are not going to read the post and still comment about it.
No, it doesn't. It never talks about Terms, nor The Distributive Law (which isn't the same thing as the Distributive Property). These are the 2 rules of Maths which make this 100% not ambiguous.
The only correct answer is
8008135.Oh i get it, if you flip that upside down it says "seiboob"
Which is French for "the boob".
1337 5|*34X 15 [)34[) |V0\/\/ 83(4(_)53 0|= 70(_).
It sure is. 13 year old me would have no trouble deciphering this, but I only got up to [)34[).
Edit: I still got it. L337 H4xor PhoR L1F3
Leet speak is dead now because of you.
Meanwhile programmers will be like, fools, clearly
2(n)is a function 😏Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma. At the end of the day it seems mathematical notation is just as flexible as any other facet of written human communication and the real answer is “make things as clear as possible and if there is ambiguity, further clarify what you are trying to communicate.”
I disagree. Without explicit direction on OOO we have to follow the operators in order.
The parentheses go first. 1+2=3
Then we have 6 ÷2 ×3
Without parentheses around (2×3) we can't do that first. So OOO would be left to right. 9.
In other words, as an engineer with half a PhD, I don't buy strong juxtaposition. That sounds more like laziness than math.
Build two cases, calculate for both, drag both case through the entirety of both problems, get two answers, make a case for both answers, end up with two hypothesis. Easy!
Now create a test to determine which case is actually correct!
Check a high school Maths textbook - even easier!
1 2 + 2 * 6 /
What's the problem?
Also, you forgot my inlaws, one of whom believes the answer is 5.
Found the reverse Pole.
It's actually 6 2 / 1 2 + *
I'm not sure if you read the post yet but I also have a short section about alternative notations which are less ambiguous or never ambiguous. RPN has the same issue as most notations that are never ambiguous namely that it's hard to read - especially for big expressions.
It's three cubits in diameter and no ne cubits around.
Therefore π is three.
Fackz.
Interesting, I didn't know about strong implicit multiplication. So I would have said the result is 9. All along my studies in France, up to my physics courses at University, all my teachers used weak implicit multiplication. Could be it's the norm in France, or they only use it in math studies at University.
In a scientific context it's actually very rare to run into that issue because divisions are mostly written as fractions which will completely mitigate the issue.
The strong implicit multiplication will only cause ambiguity after a division with inline notation. Once you use fractions the ambiguity vanishes.
In practice you also rarely see implicit multiplications between numbers but mostly between variables or variables and their coefficients.
Def not a math major (BS/PharmD), but your explanation was like seeing through a visual illusion for the first time! lol
I was always taught PEMDAS growing up, and that the MD and the AS was read left to right in an equation like above. But stating the division as a fraction completely changes my mind now about how this calculation works. I think what would happen in a calculation I use every day if the former was used.
Example: Cockcroft-Gault Equation (estimation of renal function)
(140-age)(kg) / 72(SCr) vs (140-age) X kg ➗72 X SCr
In the first eq (correct one) an 80yo patient who weighs 65kg and has an SCr ~ 1.5 = 36.11
In the latter it = 81.25 (waaay too high for an 80yo lol)
edit: calculation variable
But division and fraction aren't the same thing - the former separates terms, the latter is a single term.
The different answers for these two isn't because of / vs ➗, but because in the second one you have added extra multiplications in, thus breaking up some of the terms, and SCr has consequently been flipped from being in the denominator to being in the numerator. i.e. AK/72Scr vs. AK/72xSCr.
Yes of course, we always used fractions so there was no ambiguity. Last time I saw the division symbol must have been in primary school!
You would've done dividing by fractions in high school, which requires both. Fractions and division aren't the same thing.
I didn't know until now that I unconsciously use strong implicit multiplication (meaning that I get the answer "1"). I believe it happens more or less as a consequence of starting inside the parentheses and then working my way out.
It is a funny little bit of notational ambiguity, so it is funny that people get riled up about it.
It's not ambiguous - it's The Distributive Law. You got the correct answer, you just forgot what the rule is called (as opposed to people who forget the rule altogether).
I really hate the social media discussion about this. And the comments in the past teached me, there are two different ways of learning math in the world.
True, and it's not only about learning math but that there is actually no consensus even amongst experts, about the priority of implicit multiplications (without explicit multiplication sign). In the blog post there are a lot of things that try to show why and how that's the case.
It's not taught 2 different ways. It's taught the same around the world (the mnemonics are different but the rules are the same), there's just 2 types of people - those who remember the rules and those who don't. You'll notice students never get these questions wrong, only adults who've forgotten the rules.
Meanwhile, I'm over in the corner like
I would also add that you shouldn't be using a basic calculator to solve multi part problems. Second, I haven't seen a division sign used in a formal math class since elementary and possibly junior high. These things are almost always written as fractions which makes the logic easier to follow. The entire point of working in convention is so that results are reproducible. The real problem though is that these are not written to educate anyone. They are deliberately written to confuse so that some social media personality can make money from clicks. If someone really wants to practice math skip the click and head over to the Kahn Academy or something similar.
When I used to play WoW years ago I'd always put
-6 x 6 - 6 = -12in trade chat and they would all lose their minds. Adding that incorrect solution usually got them more riled up than having no solution.I think this speaks to why I have a total of 5 years of college and no degree.
Starting at about 7th grade, math class is taught to every single American school child as if they're going to grow up to become mathematicians. Formal definitions, proofs, long sets of rules for how you manipulate squiggles to become other squiggles that you're supposed to obey because that's what the book says.
Early my 7th grade year, my teacher wrote a long string of numbers and operators on the board, something like 6 + 4 - 7 * 8 + 3 / 9. Then told us to work this problem and then say what we came up with. This divided us into two groups: Those who hadn't learned Order of Operations on our own time who did (six plus four is ten, minus seven is three, times eight is 24, plus three is 27, divided by nine is three) Three, and who were then told we were wrong and stupid, and those who somehow had, who did (seven times eight is 56, three divided by nine is some tiny fraction...) got a very different number, and were told they were right. Terrible method of teaching, because it alienates the students who need to do the learning right off the bat. And this basically set the tone until I dropped out of college for the second time.
Yes, unfortunately there are some bad teachers around. I vividly remember the one I had in Year 10, who literally didn't care if we did well or not. I got sick for an extended period that year, and got a tutor - my Maths improved when I had the tutor (someone who actually helped me to learn the material)!
I read the whole article. I don't agree with the notation of the American Physical Society, but who am I to argue that? 😄
I started out thinking I knew how the order of operations worked and ended up with a broader view of the subject. Thank you for opening my mind a bit today. I will be more explicit in my notations from now on.
Thank you so much for taking the time. I'm also not convinced that APS's notation is a very good choice but I'm neither american nor a physisist 🤣
I'd love to see how the exceptions work that the APS added, like allowing explicit multiplications on line-breaks, if they still would do the multiplication first, but I couldn't find a single instance where somebody following the APS notation had line-break inside an expression.
I clicked on the link to see what you were talking about, and the quotes which are used in the blog aren't in there at all. i.e. I searched the whole document, not just the referenced page, and, for example, the expression "multiplication before division" isn't in there at all. On the other hand the stuff about not inserting multiplication signs into terms is 100% correct, because you are breaking up one term into two, and dropping the precedence from Terms to Multiplication, which changes the answer.
Direct quote from the article:
Yep, that's the "quote" in the blog, but if you click on the link not only is it not on page 21, it's not in there at all. i.e. the quote - if it even is a quote - is out of context.
You guys are doing it all wrong: ask chatgpt for the correct answer and paste it here. Done.
Who needs to learn or know anything really?
ChatGPT's Answer:
Chat GPT's answer for me:
I think it's funny that ChatGPT figured out 1 and 9 but has the steps completely backwards. First it points out what has high priority and then does the exact opposite, both times 🤣
ChatGPT says...
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.
My only complaint is the suggestion that engineers like to be clear. My undergrad classes included far too many things like
2 cos 2 x sin yHi, I’m stupid, is it 1+2 first, then multiple it by 2, then divide 6 by 6?
Or is it 1+2, then divide 6 by 2, then multiple?
I think it’s the first one but I’ve got no idea.
It's actually "both". There are two conventions. One is a bit more popular in science and engineering and the other one in the general population. It's actually even more complicated than that (thus the long blog post) but the most correct answer would be to point out that the implicit multiplication after the division is ambiguous. So it's not really "solvable" in that form without context.
You’d think we would’ve solve this with Einstein or Aristotle or something.
It's not a math problem, it's a communication problem. The person who wrote it down didn't make themselves clear
It's totally clear. It's a number divided by a factorised term, as per The Distributive Law and Terms.
Indeed it was already solved more than 100 years ago. The issue isn't that it's "ambiguous" - it isn't - it's that people have forgotten what they were taught (students don't get this wrong - only adults). i.e. The Distributive Law and Terms.
As if, people still can't agree if zero is a natural number either
It's the first, as per The Distributive Law and Terms. It could only ever be the second if the 6/2 was in brackets. i.e. (6/2)(1+2).
I don't remember everything, but I remember the first two operations are exponents then parentheses. Edit: wait is it the other way around?
Yes it's the other way round. Parentheses are top priority.
The full story is actually more nuanced than most people think, but the post is actually very long (about 30min) so thank you in advance if you really find the time to read it.
Forgot the algebra using fruit emoji or whatever the fuck.
Bonus points for the stuff where suddenly one of the symbols has changed and it's "supposedly" 1/2 or 2/3 etc. of a banana now, without that symbol having been defined.
This meme is specifically about the implicit multiplication because the article it links to is about that too.
But you are right there are a lot more "viral math" things than just the implicit multiplication problems 🤣
I found a few typos. In the 2nd paragraph under the section "strong feelings", you use "than" when it should be "then". More importantly, when talking about distributive properties, you say x(x+z)=xy+xz. I believe you meant x(y+z)=xy+xz.
Otherwise, I enjoyed that read. I'm embarrassed to say that I did think pemdas meant multiplication came before division, however I'm proud to say that I've unconsciously known that it's important to avoid the ambiguity by putting parentheses everywhere for example when I make formulas in spreadsheets. Which by the way, spreadsheets generally allow multiplication by juxtaposition.
Thank you so much for taking the time and reading the post. I just fixed the typos, many thanks for pointing them out.
There is nothing really to be embarrassed about and if you look at the comment sections of such viral math posts you can see that you are certainly not the only one. I think that mnemonics that use "MD" and "AS" without grouping like in "PE(MD)(AS)" are really to blame here.
An alternative would be to drop the inverse and only use say multiplication and addition as I suggested with "PEMA" but with "PEMDAS" one basically sets up students for the problem that they think that multiplication comes before division.
It’s actually fine to do multiplication before division, you just have to be sure about which numbers are intended to be included in the divisor of your fraction!
Actually it should be x(y+z)=(xy+xz), as that's exactly where a lot of people go wrong. They go from 6/2(1+2) to 6/2x3, instead of to 6/(2x3), and thus end up with the wrong answer (cos that flipped the 3 from being in the denominator to being in the numerator. i.e. instead of dividing by 3 they are now multiplying by 3, all because they removed brackets prematurely).
I just finished your article and wow! I'm definitely going to save it and share it the next time I come across another one of those viral problems. It was incredibly thorough and well researched, you clearly put a lot of energy and effort into it and it blew me away. It was really refreshing to see someone articulate themselves so passionately with supporting research. I look forward to reading more of your work!! 👏
Thank you for your kind words, really appreciate it.
It never mentions the 2 relevant rules of Maths, nor any textbooks, nor speaks to any Maths teachers. You can find all those thing here
My years out of school has made me forget about how division notation is actually supposed to work and how genuinely useless the ÷ and / symbols are outside the most basic two-number problems. And it's entirely me being dumb because I've already written problems as 6÷(2(1+2)) to account for it before. Me brain dun work right ;~;
There's no forms consensus on which one is correct. To avoid misunderstanding mathematicians use a horizontal bar.
The one that is least ambiguous is objectively more correcter.
Only if it's a fraction. If it's 2 separate terms then you use whatever your country uses for division - obelus or colon or whatever. They have to be 2 separate things, otherwise how would you write to divide by a fraction?
I guess if you wrote it out with a different annotation it would be
6
-‐--------‐--------------
2(1+2)
=
6
-‐--------‐--------------
2×3
=
6
--‐--------‐--------------
6
=1
I hate the stupid things though
Escape symbols?
Lemmy* markdown is a fuckin mess. It's way better elsewhere. & <>
Yeah connect for lemmy didn't sort the out very well.
You're more patient than me to go to that trouble! 😂 But yeah, looks good. Just one technicality (and relates to how many people arrive at the wrong answer), the 2x3 should be in brackets. Yes if you had a proper fraction bar it wouldn't matter, but that's what's missing with inline writing, and is compensated for with brackets (and brackets can't be removed unless there's only 1 term inside). In your original comment, it does indeed look like 6/(2x3), but, to illustrate the issue with what you wrote, as soon as I quoted it, it now looks like (6/2)x3 in my comment.
Lemmy interprets some symbols as formatting commands, for example putting a # at the start of a line turns it into a header:
## header
You can tell it to not do that by putting a backslash before the symbol:
\# not a header
The backslash is called the escape symbol.
Cheers mate
I guess if you wrote it out with a different annotation it would be
6
--‐--------‐-------------- 2(1+2)
= 6 --‐--------‐-------------- 2×3
= 6 --‐--------‐-------------- 6
=1
I hate the stupid things though
You lost me on the section when you started going into different calculators, but I read the rest of the post. Well written even if I ultimately disagree!
The reason imo there is ambiguity with these math problems is bad/outdated teaching. The way I was taught pemdas, you always do the left-most operations first, while otherwise still following the ordering.
Doing this for 6÷2(1+2), there is no ambiguity that the answer is 9. You do your parentheses first as always, 6÷2(3), and then since division and multiplication are equal in ordering weight, you do the division first because it's the left most operation, leaving us 3(3), which is of course 9.
If someone wrote this equation with the intention that the answer is 1, they wrote the equation wrong, simple as that.
The calculator section is actually pretty important, because it shows how there is no consensus. Sharp is especially interesting with respect to your comment because all scientific Sharp calculators say it's 1. For all the other brands for hardware calculators there are roughly 50:50 with saying 1 and 9.
So I'm not sure if you are suggesting that thousands of experts and hundreds of engineers at Casio, Texas Instruments, HP and Sharp got it wrong and you got it right?
There really is no agreed upon standard even amongst experts.
Hi, expert here, calculators have nothing to do with it. There's an agreed upon "Order of Operations" that we teach to kids, and there's a mutual agreement that it's only approximately correct. Calculators have to pick an explicit parsing algorithm, humans don't have to and so they don't. I don't look to a dictionary to tell me what I mean when I speak to another human.
Thanks for putting my thoughts into words, that's exactly why I hate math. It was supposed to be the logical one, but since it only needs to be parsed by humans it failed at even that. It's just conventions upon conventions to the point where it's notably different from one teacher/professor to the next.
I guess you can tell why I went into comp-sci (and also why I'm struggling there too)
No there isn't. I've never seen a single Year 7-8 Maths textbook that is in the slightest bit ambiguous about it. The Distributive Law has to literally always be applied (hence why it's a law). dotnet.social/@SmartmanApps/110819283738912144
The order of operations is not the same as the distributive law.
The first step in order of operations is solve brackets. The first step in solving unexpanded brackets is to expand them. i.e. The Distributive Law. i.e. the ONLY time The Distributive Law ISN'T part of order of operations is when there's no unexpanded brackets in the expression.
Used to not be. Except for Texas Instruments all the others reverted to doing it correctly now - I have no idea why Texas Instruments persists with doing it wrong. As you noted, Sharp has always done it correctly.
Yes there is. It's taught in literally every Year 7-8 Maths textbook (but apparently Texas Instruments don't care about that).
No, those companies aren't wrong, but they're not entirely right either. The answer to "6 ÷ 2(1+2)" is 1 on those calculators because that is a badly written equation and you(not literally you, to be clear) should feel bad for writing it, and the calculators can't handle it with their rigid hardcoded logic. The ones that do give the correct answer of 9 on that equation will get other equations wrong that it shouldn't be, again because the logic is hardcoded.
That doesn't change the fact that that equation worked out on paper is absolutely 9 based on modern rules of math. Calculate the parentheses first, you then have 6 ÷ 2(3). We could solve from here, but to make the point extra clear I'm going to actually expand this out to explicit multiplication. "2(3)" is the same as "2 x 3", so we can rewrite the equation as "6 ÷ 2 x 3". All operators now inarguably have equal precedence, which means the only factor left in which order to do the operations is left to right, and thus division first. The answer can only be 9.
There has apparently been historical disagreement over whether 6÷2(3) is equivalent to 6÷2x3
As a logician instead of a mathemetician, the answer is "they're both wrong because they have proven themselves ambiguous". Of course, my answer would be RPN to be a jerk or just have more parens to be a programmer
You just did division before brackets, which violates order of operations rules. 6÷2(3)=6÷(2x3)=6÷6=1
I don't see the problem actually.
==========
1+2= 3
No exponents
Nothing remains
A couple of edits (not trying to be rude but people sent to your article are going to be pedantic)
*current beliefs
This sentence needs editing: "They even split the category into two and the make a distinction between implicit multiplication with variables other implicit multiplications."
Thank you for reading the post, and thanks for pointing that out. Should be fixed and live in the next few minutes.
Update: Also fixed that sentence. Thank you so much.
I agree with your core message, that the issue is caused by bad notation. However I don't really see why you consider implicit multiplication to be the sole reason. In my mind, a/bc is equally as ambiguous as a/b*c. The symbols are not important.
You don't even consider this in your article, instead you seem to take the position that the operations are resolved from left to right. This idea probably comes from programming languages, as they commonly use this convention, but I haven't seen this defined in mathematics anywhere. I'm open to being wrong here, so if you can show me such a definition from an authoritative source (maybe ISO) I'd be thankful.
As it stands, you basically claim "the original notation is ambiguous, but with explicit × the answer is obviously nine, because my two calculators agree", even though you just discounted calculator proofs. By the way, both calculators explicitly define this left-to-right order in their documentation.
The ISO section 7.1.3 you quoted is very reasonable and succinct, and contradicts your claim that explicit multiplication sign removes ambiguity. There would be no need for this section if a left-to-right rule existed.
I'm not sure if I'd call it the "scientific" one. I'd actually say that the weak juxtaposition is just the simple one schools use because they don't want to confuse everyone. Scientist actually use both and make sure to prevent ambiguity. IMHO the main takeaway is that there is no consensus and one has to be careful to not write ambiguous expressions.
Schools don't teach "weak juxtaposition" - they teach the actual rules of Maths! As per what's in Maths textbooks. It's adults who've forgotten the rules who make up the "weak juxtaposition" rule. See Lennes.
I’m a scientist and I’ve only ever encountered strong juxtaposition in quick scribbles where everyone knows the equation already. Normally we’re very careful to use fraction notation (or parentheses) when there’s any possibility of ambiguity. I read the equation and was shocked that anyone would get an answer other than 9.
As a Maths teacher, I'm shocked whenever anyone ever gets an answer other than 1. I'm not sure how you came up with 9 when you previously said you've only ever seen strong juxtaposition? You can only get 9 with so-called "weak juxtaposition" (which is wrong).
The blog post also completely ignores what is actually taught in high school - as found in Year 7-8 Maths textbooks - which indicates how much credibility you should attach to the blog post - none.
Not sure how you came up with that conclusion. I never said anything about it being "just a blog post".
You said...
And I'm pointing out he arrived at that by ignoring what's taught in high school, which is where it's taught (not in academia). It's like saying "It's ambiguous if there's such a thing as rain" if you present weather evidence which has omitted every single rainy day that has happened. i.e. cherry-picking. Every single blog which says it's ambiguous has done the exact same thing. You can find what actually is taught in high school here
We do teach children how to solve this. It's not children who get it wrong - it's adults that get it wrong! Cos they've forgotten the rules of Maths (in this case The Distributive Law and Terms).
There are plenty of bugs that are well documented. I can't tell you the number of times that I've seen someone do something wrong, that they think is 100% right, and "carefully" document it. Then someone finds an edge case and points out the defined behavior has a bug, because the human forgot to account for something.
The other thing I'd point out that I didn't see in your blog is that I've seen many many people say they need to evaluate the 2(3) portion first because "parenthesis". No matter how many times I explain that this is a notation for multiplication, they try to claim it doesn't matter because parenthesis. screams into the void
The fact of the matter is that any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity. These viral math posts are just designed to insert ambiguity where it shouldn't be, and prey on people who can't remember middle school math.
I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher
"when in doubt" is a bit broad but left to right is a great default for operations with the same priority. There is actually a way to calculate in any order if divisions are converted to multiplications (by using the reciprocal value) and subtractions are converted to additions (by negating the value) that requires at least a little bit of math knowledge and experience so it's typically not taught until later to prevent even more confusion.
For example this: 6 / 2 * 3 can also be rewritten as 6 * 2⁻¹ * 3 and because multiplication is commutative you can now do it in any order for example like 3 * 6 * 2⁻¹
You can also "rearrange" the order without changing the meaning if you move the correct operation (left to the number) with it (should only be done with explicit multiplication)
6 / 2 * 3 into 6 * 3 / 2 (note that I moved the division with the 2)
You can even bring the two to the front. Just remember that left to the six is an "imaginary" (don't quote me ^^) multiplication. And because we can't just move "/2" to the beginning we have to insert a one (empty product - check Wikipedia) like so:
1 / 2 * 6 * 3
This also works for addition and subtraction
7 + 8 - 5
You can move them around if you take the operation left to the number with it. With addition the "imaginary" operation at the beginning is a plus sign and the implicit number you use is zero (empty sum - check Wikipedia)
8 - 5 + 7
or like this
0 - 5 + 8 + 7
because with negative numbers you can use the minus sign to indicate negative numbers you can even drop the leading zero like this
-5 + 8 + 7
That's not really possible with multiplication because "/2" is not a valid notation for "1/2"
Semi-related: something in me wants to read that as 6 / (2*3), because 6 * 3 / 2 feels like a much more 'natural' way to write it
100% related actually, since that's the actual next line of working out. i.e. you cannot remove brackets unless there is only 1 term left inside, a mistake which those who have prematurely removed brackets have made and ended up with the wrong answer (because it flips the 3 from being in the denominator to being in the numerator).
And note that it doesn't work if the multiply was an addition. e.g. 6/2+3=6 but 6+3/2=7.5. Multiplication and division are both binary operators, and you can't move them around unless you also move the term to the left with it. i.e. 6/2+3=6. 3+6/2=6.
No, to the left of the 6 is an actual plus sign, but we don't write plus signs if it's at the start of an expression. +6 and x6 aren't the same thing at all (and, since x is a binary operator, you couldn't write just x6 anyway - there would have to be a term to it's left). No expression ever starts with x6.
It's not a valid notation for multiplication either - both multiplication and division are binary operators and must be written with 2 terms.
No, that's an actual convention of Maths, to make sure people (who don't know better) obey the actual rule of left associativity.
The order of operations is not part of a holy text that must be blindly followed. If these numbers had units and we knew what quantity we were trying to solve for, there would be no argument whatsoever about what to do. This is a question that never comes up in physics because you can use dimensional analysis to check to see if you did the algebra correctly. Context matters.
No, it's in Maths textbooks, and must be... blindly followed. :-)
...it wouldn't matter at all. The order of operations comes from the very definitions of the operators themselves. e.g. 2x3 is shorthand for 2+2+2.
Hi! Nice blog post. Since you asked for feedback I'll point out the one thing I didn't really understand. You explain the difference between the calculators by showing excerpts from the manuals and you highlight that in the first manual, implicit multiplication is prioritised. But the text you underlined only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about "regular" implicit multiplication like 2(1+3). So while your photos of the calculator results are great proof that the two models use a different order of operations, to me the manuals were a bit confusing since they did not actually seem to prove your point for the example math problems you are discussing. Or maybe I missed something?
What the heck are you all fighting about? It’s BODMAS.
I'd would be great if you find the time to read the post and let me know afterwards what you think. It actually looks trivial as a problem but the situation really isn't, that's why the article is so long.
I was being facetious. I will try to find the time to read the post, but I know already that the problem isn’t trivial. It involves, above all else, human comprehension, which is a very iffy thing, to say the least.
Oh sry. I'm one of those people who are to stupid to detect sarcasm in text comments, unless it's very obvious. Probably a combination of it actually being a hard problem and me not being a native speaker.
Because it actually is.
The article was really long because there were so many stawmen in it. Had you checked a Maths textbook or asked a Maths teacher it could've been really short, but you never did either.
So what does BODMAS sound like to the other side?
samdob
They're arguing about whether Distribution is Multiplication or not. Spoiler alert: it isn't, it's Brackets.
This is a very nice piece that had so much information I did not know. Toward the top of the article I was wishing for footnotes, references or something that would indicate it was not just your opinion, but as I got further into the piece you provided so many great references. I thought the calculator manuals were particularly accessible and convincing. Thanks for a great read!
Except for any actual Maths textbooks. Try this instead.
.
That's cool and Imma let you finish but I'm not a mathematician and the answer is 9.
That's the correct answer if you follow one of the conventions. There are actually two conflicting but equally valid conventions. The blog explains the full story but this math problem is really ambiguous.
...the rules of Maths.
I read it. And I'm not a mathematician, so the answer is 9.
E: The salty mathematicians down-voting this can get fucked lmfao
Ooh now I get you, sry. True. But sadly you now know the truth and you have to be careful with the implicit multiplications on your tax forms from now on ;-)
I'm a Mathematician and the answer is 1.
I am so glad that nothing I do in life will ever cause this problem to matter to me.
The way I was taught in school, the answer is clearly 1, but I did read the blog post and I understand why that's actually ambiguous.
Fortunately, I don't have to care, so will sleep well knowing the answer is 1, and that I'm as correct as anyone else. :-p
It isn't actually ambiguous. You have remembered what you were taught in school, unlike the author of the blog post, who manages to write the whole thing without ever once checking a Maths textbook (which would reveal the only correct answer to be 1).
The fuck? I'm getting 15.
If you are not kidding, can you show your steps I can try to help you, but I can't currently think of a way how you'd end up with 15.
isn't that division sign I only saw Americans use written like this (÷) means it's a fraction? so it's 6÷2, since the divisor (or what is it called in english, the bottom half of the fraction) isn't in parenthesis, so it would be foolish to put the whole 2(1+2) down there, there's no reason for that.
so it's (6/2)*(1+2) which is 3*3 = 9.
the other way around would be 6÷(2(1+2)) if the whole expression is in the divisor and than that's 1.
tho I'm not really proficient in math, I have eventually failed it in university, but if I remember my teachers correctly, this should be the way. but again, where I live, we never use the ÷ sign, only in elementary school where we divide on paper. instead we use the fraction form, and with that, these kind of seemingly ambiguous expressions doesn't exist.
Great read! Easy for everyone to understand, but also thorough. I loved the breakdown into the calculators functionality
Interesting that Excel sees =6/2(1+2) as an invalid formula and will not calculate it (at least on mobile). =6/2*(1+2) returns 9 because it's executing the division and multiplication left to right (6/2=3*3=9).
Google Sheets (mobile) does't like it either and returns an error. =6/2*(1+2) also returns "9".
I would do the mighty parentheses first, and then the 2 that dares to touch the mighty parentheses, finally getting to the run-of-the-mill division. Hence the answer is One.
It would be 6 / 2(3) right? Which as you say would be 1, doesn't seem that hard tbh
I would be very grateful if you find the time to read the post (even if it's very long) and let me know what you think after you read it. Thank you very much in advance.
FACT CHECK 3/5
The rules are in every high school Maths textbook. The notation for your country is in your country's Maths textbooks
1+1=2 by definition (or whatever the notation is in your country). If you write 1+1=3 then that is wrong by definition
And you seem to have included most of them so far - "implicit multiplication", "weak juxtaposition", "conventions", etc.
Spoiler alert: It's always the latter
In fact what would happen is now people wouldn't know in what order to do division and subtraction, having removed them from the mnemonic (and there's absolutely no reason at all to remove them - you can do everything in the mnemonic order and it works, provided you also obey the left-to-right rule, which is there to make sure you obey left associativity)
That's not true at all. Have you not read through some of these arguments? They're all full of "Use BEDMAS!", "Use PEMDAS!", "It's PEMDAS not BEDMAS!" - quite clearly these people DID learn order of operations through the mnemonics
There's no requirement to memorise any acronym - you can always just make up your own if you find that easier! I did that a lot in university to remember things during the exam
...because a product is a Term, and to insert a x would break it into 2 Terms
A product is the result of a multiplication
Exact same reason. They are saying "don't turn 1 term into 2 terms". To put that into the words that you keep using, "don't use weak juxtaposition"
Because it would break the rule of left associativity (i.e. left to right). No-one is advocating "multiplication before division" where it would violate left to right (usually by "multiplication" they're actually referring to Terms, and yes, you literally always have to do Terms before Division)
Yes there is. Some countries use : for divide, whereas other countries use it for ratio
Name one! Give me a reference! There's nothing forbidding that in Maths (though we would more usually write it as 12/(6x2)). Again, all you have to do is obey left to right
...Terms. Same as all textbooks do now
...Terms, the already-existing rule that he apparently didn't know about (he mentions them, and products, but manages to completely miss what that actually means)
Something, something, Distributive Law (yes, some people use the wrong name, but in talking about the property, not the law, you're knocking down a strawman)
...and The Distributive Law applies to every bracketed term that has a coefficient, in this case it's 2(1+2)
And The Distributive Law has everything to do with order of operations, since solving Brackets is literally the first step!
Maybe you should've asked someone. Hint: textbooks/teachers
Here it is again, textbook references, proofs, memes, the works
Bingo! Distribution isn't Multiplication
...distribute the 2, always
It has everything to do with there being a coefficient to the brackets, the 2
...it's a factorised term, and the opposite of factorising is The Distributive Law
No, it forces distribution of the coefficient. a(b+c)=(ab+ac)
...it is a factorised term subject to The Distributive Law
They're NOT 2 separate numbers. It's a single, factorised term, in the same way that 2a is a single term, and in this case a is equal to (1+2)!
Because 2π is a single term, by definition (it's the product of a multiplication), as is r itself, so that should actually be written r=(C/2π)
Incorrect. Only the first one is a term/product (not separated by any operators) - the last 2 are multiplications, and the 2nd one is literally meaningless. Space isn't defined as meaning anything in Maths
The first is a fraction
The second is a division
The third is also a fraction
The last is a multiplication by a fraction
Creates ambiguity since space isn't defined to mean anything in Maths. Looks like a typo - was there meant to be a multiply where the space is? Or was there not meant to be a space??
By definition ab^-1^=a^1^b^-1^=(a/b)
You state that the ambiguity comes from the implicit multiplication and not the use of the obelus.
I.e. That 6 ÷ 2 x 3 is not ambiguous
What is your source for your statement that there is an accepted convention for the priority of the iinline obelus or solidus symbol?
As far as I’m aware, every style guide states that a fraction bar (preferably) or parentheses should be used to resolve the ambiguity when there are additional operators to the right of a solidus, and that an obelus should never be used.
Which therefore would make it the division expressed with an obelus that creates the ambiguity, and not the implicit multiplication.
(Rest of the post is great)
In this case it's actually the absence of sources. I couldn't find a single credible source that states that ÷ has somehow a different operator priority than / or that :
The only things there are a lot of are social media comments claiming that without any source.
My guess is that this comes from a misunderstanding that the obelus sign is forbidden in a lot of standards. But that's because it can be confused with other symbols and operations and not because the order of operations is somehow unclear.
What is your source for the priority of the / operator?
i.e. why do you say 6 / 2 * 3 is unambiguous?
Every source I’ve seen states that multiplication and division are equal priority operations. And one should clarify, either with a fraction bar (preferably) or parentheses if the order would make a difference.
Same priority operations are solved from left to right. There is not a single credible calculator that would evaluate "6 / 2 * 3" to anything else but 9.
But I challenge you to show me a calculator that says otherwise. In the blog are about 2 or 3 dozend calculators referenced by name all of them say the same thing. Instead of a calculator you can also name a single expert in the field who would say that 6 / 2 * 3 is anything but 9.
Will you accept wolfram alpha as credible source?
https://mathworld.wolfram.com/Solidus.html
Did you read the blog post? I also quoted the exact same thing.
My apologies, I wasn’t trying to spar with you friend, just trying to understand why a/b*c wouldn’t also be considered ambiguous, particularly since an author could have written a*c/b and removed any doubt.
In your blog post you also quoted ISO
You seemed to speak rather definitively that it’s only ambiguous when combined with implicit multiplication.
I agree that almost all calculators and programming languages will interpret consecutive explicit multiplications and divisions with left-to-right precedence.
But as far as I’m aware no such LTR rule has global agreement in mathematics, I was curious if you found something in your research that says otherwise.
It depends on what you mean by global agreement as there is no single source of truth but the left-to-right rule is pretty much default for multiplications/divisions and additions/subtractions. If you however have inline power notations with "^" symbol they are evaluated right-to-left. There are exceptions but those are typically well known in the industry. For example MathCad also evaluates powers from left to right, which is fairly untypical.
It's not wrong if you make clear what you are doing. You can for example in a diagram call the axis a and b, not really wrong but pretty untypical if everybody else uses x and y, so you should have at least good reasons when doing it differently.
The link references "a/bc" not "a/b*c". The first is ambiguous, the second is not.
Neither is ambiguous. #MathsIsNeverAmbiguous ab=(axb) by definition. Here it is referred to in Cajori nearly 100 years ago (1928), and literally every textbook example quoted by Lennes (1917) follows the same definition, as do all modern textbooks. Did you not notice that the blog didn't refer to any Maths textbooks? Nor asked any Maths teachers about it.
FACT CHECK 5/5
Maths teachers already know how to do Maths. Huh, who would've thought? Next thing you'll be telling me is English teachers know the rules of grammar and how to spell!
Literally NOTHING can convince a Maths teacher it's ambiguous - Maths teachers already know all the rules of Maths, and which ones you're breaking
To fact check it for the benefit of others
At this point I hope you understand why it isn't ambiguous. Tip: next time check some Maths textbooks or ask a Maths teacher
Neither of them is a thing
Facts you mean. The rules of Maths are facts
There absolutely is! You just chose not to ask any experts about it
...they are all solvable by following the rules of Maths
There is - The Distributive Law and Terms
No, BIDMAS and left to right is the full story
No, post a link to this order of operations thread index - it has textbook references, proofs, memes, worked examples, the works!
FACT CHECK 4/5
There's absolutely nothing wrong with doing that. The order of operations rules have everything covered. Anything which follows an operator is a separate term. Anything which has a fraction bar or brackets is a single term
Because they don't come with order of operations built-in - the programmer has to implement it (which is why so many e-calculators are wrong)
Microsoft has 3 different software packages which get order of operations wrong in 3 different ways, so I wouldn't be using them as an example! There are multiple rules of Maths they don't obey (like always rounding up 0.5)
This is wrong in so many ways!
From those 7 points, we can see this is not a real Maths problem. You deliberately made it ambiguous (didn't say what any of the pronumerals are) so you could say "Look! Maths is ambiguous!". In other words, this is a strawman. If you really think Maths is ambiguous, then why didn't you use a real Maths example to show that? Spoiler alert: #MathsIsNeverAmbiguous hence why you don't have a real example to illustrate ambiguity
No they can't. See previous points. If there is a function, then you have to define what it is. e.g. f(x)=x². If no function has been defined, then f is the pronumeral f of the factorised term f(x), not a function. And also, if there was a function defined, you wouldn't use f as a pronumeral as well! You have the whole rest of the alphabet left to use. See my point about we teach appropriate choice of pronumerals
No, it really doesn't. You just literally made up some examples which go against the rules of Maths then claimed "Look! Maths is ambiguous!". No, it isn't - the rules of Maths make sure it's never ambiguous
Which is exactly what calculators do! If you type in something invalid (say you were missing a bracket), it would say "syntax error" or something similar
Are you saying they shouldn't be allowed to enter factorised terms? If so, why?
We already do
In what way is 6/2(1+2) either convoluted or hard to read? It's a term divided by a factorised term - simple
In other words, follow the rules of Maths.
Spoiler alert: they're not
e.g. fx=f(x), which I already addressed. It's either been defined as a function or as pronumerals, therefore nothing ambiguous
No, it's not. |a|b|c| is the absolute value of a, times b, times the absolute value of c... which you would just write as b|ac|. Unlike brackets you can't have nested absolute values, so the absolute value of (a times the absolute value of b times c) would make no sense, especially since it's the EXACT same answer as |abc| anyway!
Left associativity. i.e. an exponent is associated with the term to its left - solve exponents right to left
People saying "I don't know how to interpret this" doesn't mean it's ambiguous, nor that it isn't defined. It just means, you know, they need to look it up (or ask a Maths teacher)! If someone says "I don't know what the word 'cat' means", you don't suddenly start running around saying "The word 'cat' is ambiguous! The word 'cat' is ambiguous!" - you just tell them to look it up in a dictionary. In the case of Maths, you look it up in a Maths textbook
...and any of them which contradict any of the rules of Maths are demonstrably wrong
...and Maths teachers know that both of them are made-up and not real things in Maths
Nope. The mnemonics plus left to right covers everything you need to know about it
...because it's not a real thing
...or because they're a high school Maths teacher and know all the rules of Maths
Yes it can...
Forgotten rules of Maths - The Distributive Law (e.g. a(b+c)=(ab+ac)) applies to all bracketed Terms, and Terms are separated by operators and joined by grouping symbols
Bam! Done! Explained in a quick comment
FACT CHECK 2/5
...and yet still a bug (I saw at least one other person point this out to you)
A few years ago, there was a Microsoft feature intended for people in China, but people who weren't in China were getting that behaviour. i.e. a bug. It was documented and a deliberate design choice for people in China, but if you weren't in China then it's a bug. Just documenting a design choice doesn't mean bugs don't happen. A calculator giving a wrong answer is a bug
Based on the comments in the above video, the opposite is true - this problem first arose in '96
So the person programming it is far more likely to need to check their Maths first - bingo!
...and some that use both! i.e. some follow Terms but not The Distributive Law. As I said to begin with, these are 2 DIFFERENT rules, and you can't just lump them together as one
Which is correct, as per Terms
What you mean is they evaluate it as 1/2xX, since 1/2X and 1/(2X) are the same thing
No, not necessary, since 2a=(2xa) by definition, alluded to in Cajori in 1928...
...follow all the rules of Maths, always. There's something to be said for making sure you're doing it right. :-)
...and they will actually remove brackets I have put in and replace them with their own ("hi" to all the people who say you can fix any calculator by "just add more brackets" - Google doesn't CARE what brackets you've added!)
It's not, because a ÷ isn't a fraction bar. They're joining 2 terms into one and thus sometimes changing the answer
It's not that they don't allow it, it's that it's not provided with the language by default in the first place! Most languages only provide you with some numbers, operators, and a few functions (like round), and it's up to the programmer to implement the rest. Welcome to why there are so many wrong e-calculators
...which is a red flag to not use that calculator!
I'm not sure it does. I'd have to switch on "strong juxtaposition" (the only kind there is) and see what else has been disobeyed in Maths. e.g. Google removing my brackets and adding different ones
I find any exceptions to following the rules of Maths surprising! No, you can't just make up your own rules
a/bc=a/(bc) in every textbook
Welcome to "we're gonna add brackets to what you typed in and change the answer"
...then that means it's not "multiplication" - it's Terms and/or The Distributive Law. The "M" in the mnemonics refers literally to multiplication signs, nothing else
Yep, and The Distributive Law and Factorising are the inverse of each other
...and Brackets is always first, so in this case it doesn't even matter
Yes they do - mnemonics represent the actual order of operations rules
No, they won't. Year 8 is the last time order of operations is taught, and they have been taught everything they need to know about it by then
...and yet have you not noticed that teenagers almost never get this wrong - only adults do
...is a totally valid thing to do. The problem is people classifying Distribution (Brackets/Parentheses with a coefficient) as "Multiplication", when there's literally no multiplication sign
No they don't. Maths is universal
It's all based on definitions and proofs, which are immutable
You can find them in any high school textbook in your country (notation varies by country, but the rules don't)
"implicit multiplication" doesn't appear in any Maths textbooks
Yes it is clear (as I think I saw someone already point out here)
I saw the man with the telescope - the man has the telescope
I saw the man, with the telescope - I saw the man through a telescope
I saw the man through the telescope - I saw the man through a telescope
But there are proofs! (There you go again with the "there is no..." red flag) Order of operations proof
Nope it's bedmas since everything is brackets
Sorry but I don't follow. Did you read the blog post?
Those hoity toity types with their parentheses don't know what it is to struggle
FACT CHECK 1/5
No, you are. You've ignored multiple rules of Maths, as we'll see...
Except it's not ambiguous at all
...and an entire subset of those people are high school Maths teachers!
A change to the rules of Maths that's not in any textbooks yet, and somehow no teachers have been told about it yet either
I can do something for you though - fact-check your blog
There's no "belief" when it comes to rules of Maths - they are facts (some by definition, some by proof)
#MathsIsNeverAmbiguous
There's no such thing as "implicit multiplication". You won't find that term used anywhere in any Maths textbook. People who use that term are usually referring to Terms, The Distributive Law, or most commonly both! #DontForgetDistribution
Nope. It's a valid notation for a factorised Term. e.g. 2a+2b=2(a+b). And the reverse process to factorising is The Distributive Law. i.e. 2(a+b)=(2a+2b).
The only type of multiplication there is is explicit. Neither Terms nor The Distributive Law is classed as "multiplication"
There is a single, standard, order of operations rules
Also, see my thread about people who say there is no evidence/proof/convention - it almost always ends up there actually is, but they didn't look (or didn't want you to look)
...they have forgotten about Terms and/or The Distributive Law, and are trying to treat a Term as though it's a "multiplication", and it's not. More soon
Let me paraphrase - people disagree about made-up rule
There's no such thing - there's either juxtaposition or not, and if there is it's either Terms or The Distributive Law
...factorised term after that
There's no ambiguity...
multiplication sign - multiplication
brackets with no multiplication sign (i.e. a coefficient) - The Distributive Law
no multiplication sign and no brackets - Terms (also called products by some. e.g. Lennes)
Why didn't you ask a teacher before writing your blog? Maths tests are only ever ambiguous if there's been a typo. If there's no typo's then there's a right answer and wrong answers. If you think the question is ambiguous then you've not studied enough
This question already is clear. It's division, NOT a fraction. They are NOT the same thing! Terms are separated by operators and joined by grouping symbols. 1÷2 is 2 terms, ½ is 1 term
BTW here is what happened when someone asked a German Maths teacher
You should literally NEVER use "weak juxtaposition" - it contravenes the rules of Maths (Terms and The Distributive Law)
...and high school, where it's first taught
If that was what was meant then that's what would've been written - the 6 and 2 have been joined together to make a single term, and elevated to the precedence of Brackets rather than Division
You should know, without being told, to follow the rules of Maths when solving it. Voila! No ambiguity
It stirs up drama because many adults have forgotten the rules of Maths (you'll find students get this right, because they still remember)
No, you just put the cart before the horse - the problem existing in the first place (programmers not brushing up on their Maths first) is why some calculators do it wrong
Yes, we use / to mean divide with computers (since there is no ÷ on the keyboard), which you therefore need to put into brackets if it's a fraction (since there's no fraction bar on the keyboard either)
Nope. See previous comment.
Because programmers didn't check their Maths first, some calculators give wrong answers
According to this video mostly not these days (based on her comments, there's only Texas Instruments which isn't obeying both Terms and The Distributive Law, which she refers to as "PEJMDAS" - she didn't have a manual for the HP calcs). i.e. some manufacturers who were doing it wrong have switched back to doing it correctly
P.S. she makes the same mistake as you, and suggests showing her video to teachers instead of just asking a teacher in the first place herself (she's suggesting to add something to teaching which we already do teach. i.e. ab=(axb)).
ANY calculator which doesn't obey all the rules of Maths is wrong!
So a calculator, which has a specific purpose of solving Maths expressions, giving a wrong answer to a Maths expression isn't "unintended behaviour"? Do go on
Starting a new comment thread (I gave up on reading all of them). I'm a high school Maths teacher/tutor. You can read my Mastodon thread about it at Order of operations thread index (I'm giving you the link to the thread index so you can just jump around whichever parts you want to read without having to read the whole thing). Includes Maths textbooks, historical references, proofs, memes, the works.
And for all the people quoting university people, this topic (order of operations) is not taught at university - it is taught in high school. Why would you listen to someone who doesn't teach the topic? (have you not wondered why they never quote Maths textbooks?)
#DontForgetDistribution #MathsIsNeverAmbiguous
Honestly, I do disagree that the question is ambiguous. The lack of parenthetical separation is itself a choice that informs order of operations. If the answer was meant to be 9, then the 6/2 would be isolated in parenthesis.
It's covered in the blog, but this is likely due to a bias towards Strong Juxtaposition rules for parentheses rather than Weak. It's common for those who learned math into advanced algebra/ beginning Calc and beyond, since that's the usual method for higher math education. But it isn't "correct", it's one of two standard ways of doing it. The ambiguity in the question is intentional and pervasive.
My argument is specifically that using no separation shows intent for which way to interpret and should not default to weak juxtaposition.
Choosing not to use (6/2)(1+2) implies to me to use the only other interpretation.
There's also the difference between 6/2(1+2) and 6/2*(1+2). I think the post has a point for the latter, but not the former.
I originally had the same reasoning but came to the opposite conclusion. Multiplication and division have the same precedence, so I read the operations from left to right unless noted otherwise with parentheses. Thus:
6/2=3
3(1+2)=9
For me to read the whole of 2(1+2) as the denominator in a fraction I would expect it to be isolated in parentheses: 6/(2(1+2)).
Reading the blog post, I understand the ambiguity now, but i’m still fascinated that we had the same criticism (no parentheses implies intent) but had opposite conclusions.
You just did division before brackets, which goes against order of operations rules.
You just need to know The Distributive Law and Terms.
Read the linked article
The linked article is wrong. Read this - has, you know, actual Maths textbook references in it, unlike the article.
I don't know what you want, man. The blog's goal is to describe the problem and why it comes about and your response is "Following my logic, there is no confusion!" when there clearly is confusion in the wider world here. The blog does a good job of narrowing down why there's confusion, you're response doesn't add anything or refute anything. It's just... you bragging? I'm not certain what your point is.
None of this has a point. We're talking over a shitpost rant about common use of math symbols. Even the conclusion boils down to it being a context dependent matter of preference. I'm just disagreeing that the original question as posed should be interpreted with weak juxtaposition.
That's because the actual rules of Maths have all been followed, including The Distributive Law and Terms.
Amongst people who don't remember The Distributive Law and Terms.
The blog ignores The Distributive Law and Terms. Notice the complete lack of Maths textbook references in it?
It is correct - it's The Distributive Law.
There's only 1 way - the "other way" was made up by people who don't remember The Distributive Law and/or Terms (more likely both), and very much goes against the standards.
...zero.
Did you read the blog post?
Hooray! Correct! Anyone who downvoted or disagrees with this needs to read this instead. Includes actual Maths textbooks references.
Who gives a shit about order of operations. In a real world scenario you'll know what to do
This is why you're a confused possum.
Unfortunately, demonstrably, a lot of people don't know what to do.
It's not ambiguous, it's just that correctly parsing the expression requires more precise application of the order of operations than is typical. It's unclear, sure. Implicit multiplication having higher precedence is intuitive, sure, but not part of the standard as-written order of operations.
I'd really like to know if and how your view on that matter would change once you read the full post. I know it's very long and a lot of people won't read it because they "already know" the answer but I'm pretty sure it would shift your perception at least a bit if you find the time to read it.
My opinion hasn't changed. The standard order of operations is as well defined as a notational convention can be. It's not necessarily followed strictly in practice, but it's easier to view such examples as normal deviation from the rules instead of an implicit disagreement about the rules themselves. For example, I know how to "properly" capitalize my sentences too, and I intentionally do it "wrong" all the time. To an outsider claiming my capitalization is incorrect, I don't say "I am using a different standard," I just say "Yes, I know, I don't care." This is simpler because it accepts the common knowledge of the "normal" rules and communicates a specific intent to deviate. The alternative is to try to invent a new set of ad hoc rules that justify my side, and explain why these rules are equally valid to the ones we both know and understand.
What is the correct answer according to the convention you follow?
I have a masters in math, please do not condescend. I'm fully aware of both interpretations and your overall point and I've explained my response.
I still don't see a number ;-) but you can take a look at the meme to see other people with math degrees shouting at each other.
Sorry your article wasn't as interesting as you hoped.
The difference is that there are two sets of rules already in use by large groups of people, so which do you consider correct?
There's only 1 set of rules, and 2 sets of people - those who follow the rules and those who don't.
There aren't.
They weren't asking you if there are two sets of rules, we're in a thread that's basically all qbout the Weak vs. Strong juxtaposition debate, they asked you which you consider correct.
Giving the answer to a question they didn't ask to avoid the one they did is immature.
Ah yes, simply "answer the question with an incorrect premise instead of refuting the premise." When did you stop beating your wife?
That's not what they asked me. I have no problem answering questions that are asked in good faith.
I can't have stopped because I never started, because I'm not even married... See, even I can answer your bad faith question better than you answered the one @onion asked you.
But I will give it to you that my comment should've stipulated avoiding reasonable questions.
However I still think you need your eyes checked, as the end of this comment by @onion is very clearly a question asking you WHICH ruleset you consider correct.
Unless you're refusing the notion of multiplication by juxtaposition entirely, then you must be on one side of this or the other.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
A well-defined order of operations shouldn't have a hole that big.
Also, @wischi asking you to give the answer as defined by your convention isn't condescending, they're asking you to put your money where your mouth is...
Your response certainly felt condescending though, especially since your "explanation" was essentially that anyone who disagrees with the convention you follow is wrong and should feel stupid, and that you needn't even consider it.
They didn't - neither of them is a rule of Maths.
There aren't two different sets of rules. There's the simple model that's commonly understood and taught to kids, and there's the real world where you have context and the dynamics of a conversation and years of experience with communication. One is well defined, the other isn't.
Them asking me to solve the arithmetic problem is condescending, yes.
My response didn't say "anyone who disagrees with the convention is stupid." Here's condescension for you: please don't make your reading level my problem. What I said was, there's an unambiguous way to parse the expression according to the commonly understood order of operations, but it is atypical to pay that much attention to the order of operations in practice. If you think that's a value judgment, that's on you-- I was very clear in my example about capitalization, "strictly adhering to the conventional order of operations" is something reasonable people often just don't care about.
And that simple model, well-defined model didn't properly account for juxtaposition, which is how different fields have ended up with two different ways of interpreting it, i.e. strong vs. weak juxtaposition.
In the real world you simply wouldn't write any equation out in such a way as to allow misinterpretation like this, but that's ignoring the elephant in the room...
Which is that the reason viral problems like this still come about and why @wischi went through the effort of writing a rather detailed blog on this is because the order of operations most people are taught doesn't cover juxtaposition.
Considering your degree specialisation is in solving arithmetic problems, I don't see the issue with them asking you to put your money where your mouth is and spit out a number if it's so easy.
Ironic that you tell me to check my reading comprehension right after you misquote me, but nonetheless that is the impression your responses have given off - and you haven't done anything so far to dispel that impression.
Yes, and the question everyone is asking you is what is that unambiguous way? Which side of weak or strong juxtaposition do you come out on?
The value judgement was actually more to do with your choice of example, and how you applied that example to this debate. It gave me the distinct impression that you view this debate as not worth having, as anybody who does juxtaposition differently from you is wrong out the gate - and again, your further responses only reinforce my impression of you.
The order of operations rules do cover it. Did you not notice that the OP never referenced a single Maths textbook? Because, had that been done, the whole house of cards would've fallen down. See my Fact Check posts doing exactly that.
No, that's just not what happened. "Strong juxtaposition," while well-defined, is a post-hoc rationalization. Meaning in particular that people who believe that this expression is best interpreted with "strong juxtaposition" don't really believe in "strong juxtaposition" as a rule. What they really believe is that communication is subtle and context dependent, and the traditional order of operations is not comprehensive enough to describe how people really communicate. And that's correct.
My degree specialization is in algebraic topology.
The issue is that this question disregards and undermines my point and asks me to pick a side, arbitrarily, that (as I've already explained) I don't actually believe in.
I didn't misread, you're in denial.
Hopefully by this point in the comment you understand that I don't believe the question makes sense.
Again, that's your fault-- you've clearly misinterpreted what I said. If I didn't think this conversation was worth having I wouldn't be responding to you.
I think you're putting the cart before the horse here - there is definitely a communication issue around juxtaposition, but you're acting as though strong juxtaposition is some kind of social commentary on the standard order of operations rather than the reality that it is an interpretation that arose due to ambiguity, just as weak juxtaposition did.
If it were people just trying to make a point, then why would it be so widely used and most scientific calculators are designed to use it, or allow its use. This debate exists because so many people ascribe to one or the other without thinking.
One - that does sound kind of cool
Two - You still have a mathematics degree do you not? You said this was an easy "unambiguous" problem to solve, so I don't see how you're prohibited from solving it...
God saying stuff like that, you sound just like an enlightened centrist...
Anyways, even if you don't want to comment on the strong vs. weak juxtaposition debate, unless you simply intend on never solving any equation with implicit multiplication by juxtaposition ever again, then you must have a way of interpreting it.
That is what you're being asked to disclose, since you seem to be very certain that there is a correct way of resolving this. You've brought the question upon yourself.
If you don't want to take a side, simply saying the rules are ambiguous and technically both positions are correct depending on what field you're in is also a valid position...
But denying the problem all together is not productive.
In the first place I don't think you've proven me wrong. As far as I can tell your comments still boil down to that you think the whole debate is wrong, and that engaging in the debate is dumb.
But I can say for certain that you either misread or deliberately misconstrued at least part of my reply, because when responding to me you removed the "you follow" from it, which changes the interpretation.
If you think that wasn't what I said, feel free to go back and look.
I understand you don't believe the question makes sense, you've said that enough times...
But I'll just refer you to my earlier comment - unless you intend on never solving any equation involving implicit multiplication ever again, then you must ascribe to one way or the other of resolving it.
Then tell me how I've misinterpreted what you said, because I stick by what I said as far as your example goes.
Your choice of example is not only a much more clear cut issue, being that most kids are taught by primary school (or the US equivalent) how and where to capitalise their letters, and to me it also demonstrates that you've not understood that the whole reason this debate is a thing is directly because there's no "wrong way" of doing this.
I understand you see this conversation with me as worth having, but I suspect this is more to do with wanting to resolve this conversation in your favour than it is to do with the underlying debate.
This is not a math problem but a calculator engineering problem. Some solve the sub operations from right to left while other do it from left to right.
It's not really a calculator engineering problem. If you don't have time to read the entire blog you should definitely check out the section "But my calculator says...". It's actually about order of operations regarding implicit multiplication.
It's not ambiguos nor an communication problem, it's basic Math
You should read the part about WolframAlpha in the blog.
https://www.wolframalpha.com/input?i=6%2Fxy+where+x%3D2%2C+y%3D3
https://www.wolframalpha.com/input?i=6%2Fx*%28y%29+where+x%3D2%2C+y%3D3
Heyoooooo
Now you changed it to an explicit multiplication. The ambiguity only comes from the implicit multiplication after a division, that's when the interpretation can be ambiguous. That's what the blog post really is about.
Both of those screenshots, the input is a fraction, thereby removing the ambiguity. But when you use the division symbol, an ambiguity arises. This is why you should never, for any reason, use a division symbol.
...he literally used the
÷operator in the top screenshot. WolframAlpha interprets it as synonymous with/.When putting in ambiguous inputs to WolframAlpha, it does its best to interpret it so that it's can give an answer, and it shows you underneath how it interpreted it. That doesn't mean there wasn't any ambiguity to begin with.
Right. I'm saying both / and ÷ are ambiguous in that context. WA interprets both symbols as having equivalent meaning.
The wrong meaning. It interprets them both as a fraction bar, thus giving the wrong answer.
Division doesn't mean fraction. Division is 2 terms, a fraction is 1 term. Terms are separated by operators and joined by grouping symbols. If you change the division to a fraction you change the number of terms and change the answer (and you also would've just done division before brackets, which violates the order of operations rules).
And WolframAlpha did division before brackets (turned 6/2 into a fraction, thus making it a single term instead of separate terms, all before doing brackets), thus violating the order of operations rules.