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 :)
If you are so sure that you are right and already “know it all”, why bother and even read this? There is no comment section to argue.
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🤣 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
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 🤣
Nope it’s bedmas since everything is brackets
@wischi “Funny enough all the examples that N.J. Lennes list in his letter use implicit multiplications and thus his rule could be replaced by the strong juxtaposition”.
Weird they didn’t need two made-up terms to get it right 100 years ago.
Indeed Duncan. :-)
his rule could be replaced by the strong juxtaposition
“strong juxtaposition” already existed even then in Terms (which Lennes called Terms/Products, but somehow missed the implication of that) and The Distributive Law, so his rule was never adopted because it was never needed - it was just Lennes #LoudlyNotUnderstandingThings (like Terms, which by his own admission was in all the textbooks). 1917 (ii) - Lennes’ letter (Terms and operators)
In other words…
Funny enough all the examples that N.J. Lennes list in his letter use
…Terms/Products., as we do today in modern high school Maths textbooks (but we just use Terms in this context, not Products).
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.
Hooray! Correct! Anyone who downvoted or disagrees with this needs to read this instead. Includes actual Maths textbooks references.
Did you read the blog post?
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 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.
your response is “Following my logic, there is no confusion!”
That’s because the actual rules of Maths have all been followed, including The Distributive Law and Terms.
there clearly is confusion in the wider world here
Amongst people who don’t remember The Distributive Law and Terms.
The blog does a good job of narrowing down why there’s confusion
The blog ignores The Distributive Law and Terms. Notice the complete lack of Maths textbook references in it?
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.
6/2=3
3(1+2)=9
You just did division before brackets, which goes against order of operations rules.
For me to read the whole of 2(1+2) as the denominator in a fraction
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.
But it isn’t “correct”
It is correct - it’s The Distributive Law.
it’s one of two standard ways of doing it.
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.
The ambiguity in the question is
…zero.
What if the real answer is the friends we made along the way?
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! 😂
FACT CHECK 1/5
If you are sure the answer is one… you are wrong
No, you are. You’ve ignored multiple rules of Maths, as we’ll see…
it’s (intentionally!) written in an ambiguous way
Except it’s not ambiguous at all
There are quite a few people who are certain(!) that their result is the only correct answer
…and an entire subset of those people are high school Maths teachers!
What kind of evidence/information would it take to convince you, that you are wrong
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
If there is nothing that would change your mind, then I’m sorry I can’t do anything for you.
I can do something for you though - fact-check your blog
things that contradict your current beliefs.
There’s no “belief” when it comes to rules of Maths - they are facts (some by definition, some by proof)
How can math be ambiguous?
#MathsIsNeverAmbiguous
operator priority with “implied multiplication by juxtaposition”
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
This is a valid notation for a multiplication
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).
but the order of operations it’s not well defined with respect to regular explicit multiplication
The only type of multiplication there is is explicit. Neither Terms nor The Distributive Law is classed as “multiplication”
There is no single clear norm or convention
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)
The reason why so many people disagree is that
…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
conflicting conventions about the order of operations for implied multiplication
Let me paraphrase - people disagree about made-up rule
Weak juxtaposition
There’s no such thing - there’s either juxtaposition or not, and if there is it’s either Terms or The Distributive Law
construct “viral math problems” by writing a single-line expression (without a fraction) with a division first and a
…factorised term after that
Note how none of them use a regular multiplication sign, but implicit multiplication to trigger the ambiguity.
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)
If it’s a school test, ask you teacher
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
maybe they can write it as a fraction to make it clear what they meant
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 probably stick to the weak juxtaposition convention
You should literally NEVER use “weak juxtaposition” - it contravenes the rules of Maths (Terms and The Distributive Law)
strong juxtaposition is pretty common in academic circles
…and high school, where it’s first taught
(6/2)(1+2)=9
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
written in an ambiguous way without telling you what they meant or which convention to follow
You should know, without being told, to follow the rules of Maths when solving it. Voila! No ambiguity
to stir up drama
It stirs up drama because many adults have forgotten the rules of Maths (you’ll find students get this right, because they still remember)
Calculators are actually one of the reasons why this problem even exists in the first place
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
“line-based” text, it led to the development of various in-line notations
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)
With most in-line notations there are some situations with conflicting conventions
Nope. See previous comment.
different manufacturers use different conventions
Because programmers didn’t check their Maths first, some calculators give wrong answers
More often than not even the same manufacturer uses different conventions
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)).
none of those two calculators is “wrong”
ANY calculator which doesn’t obey all the rules of Maths is wrong!
Bugs are – by definition – unintended behaviour. That is not the case here
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
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What the heck are you all fighting about? It’s BODMAS.
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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.
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.
It actually looks trivial as a problem
Because it actually is.
that’s why the article is so long
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.
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.
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
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Escape symbols?
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6⁄2(1+2) ⇒ 6⁄2*3 ⇒ 6⁄6 ⇒ 1
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.
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?
only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about “regular” implicit multiplication like 2(1+3)
That was a very astute observation you made there! The fact is, for the very reason you stated, there is in fact no such thing as “implicit multiplication” - it is a term which has been made up by people who have forgotten Terms (the first thing you mentioned) and The Distributive Law (the second thing you mentioned). As you’ve noted., these are 2 different rules, and lumping them together as one brings exactly the disastrous results you might expect from lumping different 2 rules together as one…
See here for explanation of all the various rules, including textbook references and proofs.
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.
FACT CHECK 5/5
most people just dismiss that, because they “already know” the answer
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!
and a two-sentence comment can’t convince them how and why it’s ambiguous
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
Why read something if you have nothing to learn about the topic that’s so simple that you know for a fact that you are right
To fact check it for the benefit of others
At this point I hope you understand how and why the original problem is ambiguous
At this point I hope you understand why it isn’t ambiguous. Tip: next time check some Maths textbooks or ask a Maths teacher
that one of the two shouldn’t even be a thing
Neither of them is a thing
not everybody shares your opinion and preferences
Facts you mean. The rules of Maths are facts
There is no mathematically true
There absolutely is! You just chose not to ask any experts about it
the most important thing with this “viral math” expressions is to recognize that
…they are all solvable by following the rules of Maths
One could argue that there should also be a strong connection between coefficients and variables (like in r=C/2π)
There is - The Distributive Law and Terms
it’s fine to stick to “BIDMAS” in school but be aware that that’s not the full story
No, BIDMAS and left to right is the full story
If you encounter such discussions in the wild you could just post a link to this page
No, post a link to this order of operations thread index - it has textbook references, proofs, memes, worked examples, the works!
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
I’m curious if you actually read the whole (admittedly long) page linked in this post, or did you stop after realizing that it was saying something you found disagreeable?
I’m a high school Maths teacher/tutor
What will you tell your students if they show you two different models of calculator, from the same company, where the same sequence of buttons on each produces a different result than on the other, and the user manuals for each explain clearly why they’re doing what they are? “One of these calculators is just objectively wrong, trust me on this, #MathsIsNeverAmbiguous” ?
The truth is that there are many different math notations which often do lead to ambiguities.
In the case of the notation you’re dismissing in your (hilarious!) meme here, well, outside of anglophone high schools, people don’t often encounter the obelus notation for division at all except for as a button on calculators. And there its meaning is ambiguous (as clearly explained in OP’s link).
Check out some of the other things which the “÷” symbol can mean in math!
#MathNotationsAreOftenAmbiguous
did you stop after realizing that it was saying something you found disagreeable
I stopped when he said it was ambiguous (it’s not, as per the rules of Maths), then scanned the rest to see if there were any Maths textbook references, and there wasn’t (as expected). Just another wrong blog.
What will you tell your students if they show you two different models of calculator, from the same company
Has literally never happened. Texas Instruments is the only brand who continues to do it wrong (and it’s right there in their manual why) - all the other brands who were doing it wrong have reverted back to doing it correctly (there’s a Youtube video about this somewhere). I have a Sharp calculator (who have literally always done it correctly) and most of my students have Casio, so it’s never been an issue.
trust me on this
I don’t ask them to trust me - I’m a Maths teacher, I teach them the rules of Maths. From there they can see for themselves which calculators are wrong and why. Our job as teachers is for our students to eventually not need us anymore and work things out for themselves.
The truth is that there are many different math notations which often do lead to ambiguities
Not within any region there isn’t. e.g. European countries who use a comma instead of a decimal point. If you’re in one of those countries it’s a comma, if you’re not then it’s a decimal point.
people don’t often encounter the obelus notation for division at all
In Australia it’s the only thing we ever use, and from what I’ve seen also the U.K. (every U.K. textbook I’ve seen uses it).
Check out some of the other things which the “÷” symbol can mean in math!
Go back and read it again and you’ll see all of those examples are worded in the past tense, except for ISO, and all ISO has said is “don’t use it”, for reasons which haven’t been specified, and in any case everyone in a Maths-related position is clearly ignoring them anyway (as you would. I’ve seen them over-reach in Computer Science as well, where they also get ignored by people in the industry).
Has literally never happened. Texas Instruments is the only brand who continues to do it wrong […] all the other brands who were doing it wrong have reverted
Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today, and then also admit you know that it did happen with some other brands in the past?
But, if you had read the linked post before writing numerous comments about it, you’d see that it documents that the ambiguity actually exists among both old and currently shipping models from TI, HP, Casio, and Canon, today, and that both behaviors are intentional and documented.
There is no bug; none of these calculators is “wrong”.
The truth is that there are many different math notations which often do lead to ambiguities
Not within any region there isn’t.
Ok, this is the funniest thing I’ve read so far today, but if this is what you are teaching high school students it is also rather sad because you are doing them a disservice by teaching them that there is no ambiguity where there actually is.
If OP’s blog post is too long for you (it is quite long) i recommend reading this one instead: The PEMDAS Paradox.
In Australia it’s the only thing we ever use, and from what I’ve seen also the U.K. (every U.K. textbook I’ve seen uses it).
By “we” do you mean high school teachers, or Australian society beyond high school? Because, I’m pretty sure the latter isn’t true, and I’m skeptical of the former. I thought generally the ÷ symbol mostly stops being used (except as a calculator button) even before high school, basically as soon as fractions are taught. Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?
Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today
You asked me what I do if my students show me 2 different answers what do I tell them, and I told you that has never happened. None of my students have ever had one of the calculators which does it wrong.
that both behaviors are intentional and documented
Correct. I already noted earlier (maybe with someone else) that the TI calculator manual says that they obey the Primary School order of operations, which doesn’t work with High School order of operations. i.e. when the brackets have a coefficient. The TI calculator will give a correct answer for 6/(1+2) and 6/2x(1+2), but gives a wrong answer for 6/2(1+2), and it’s in their manual why. I saw one Youtuber who was showing the manual scroll right past it! It was right there on screen why it does it wrong and she just scrolled down from there without even looking at it!
none of these calculators is “wrong”.
Any calculator which fails to obey The Distributive Law is wrong. It is disobeying a rule of Maths.
there is no ambiguity where there actually is.
There actually isn’t. We use decimal points (not commas like some European countries), the obelus (not colon like some European countries), etc., so no, there is never any ambiguity. And the expression in question here follows those same notations (it has an obelus, not a colon), so still no ambiguity.
i recommend reading this one instead: The PEMDAS Paradox
Yes, I’ve read that one before. Makes the exact same mistakes. Claims it’s ambiguous while at the same time completely ignoring The Distributive Law and Terms. I’ll even point out a specific thing (of many) where they miss the point…
So the disagreement distills down to this: Does it feel like a(b) should always be interchangeable with axb? Or does it feel like a(b) should always be interchangeable with (ab)? You can’t say both.
ab=(axb) by definition. It’s in Cajori, it’s in today’s Maths textbooks. So a(b) isn’t interchangeable with axb, it’s only interchangeable with (axb) (or (ab) or ab). That’s one of the most common mistakes I see. You can’t remove brackets if there’s still more than 1 term left inside, but many people do and end up with a wrong answer.
By “we” do you mean high school teachers, or Australian society beyond high school?
I said “In Australia” (not in Australian high school), so I mean all of Australia.
Because, I’m pretty sure the latter isn’t true
Definitely is. I have never seen anyone here ever use a colon to mean divide. It’s only ever used for a ratio.
Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?
All my textbooks use both. Did you read my thread? If you use a fraction bar then that is a single term. If you use an obelus (or colon if you’re in a country which uses colon for division) then that is 2 terms. I covered all of that in my thread.
EDITED TO ADD: If you don’t use both then how do you write to divide by a fraction?
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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.
FACT CHECK 2/5
The behaviour is intended and even carefully documented in the manual
…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
weak juxtaposition is only used by old calculators
Based on the comments in the above video, the opposite is true - this problem first arose in '96
because they are scientific calculators.
So the person programming it is far more likely to need to check their Maths first - bingo!
TI (Texas Instruments) also has some calculators that use strong juxtaposition and some products that use weak juxtaposition
…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
evaluate 1/2X as 1/(2X)
Which is correct, as per Terms
while other products may evaluate the same expression as 1/2X from left to right
What you mean is they evaluate it as 1/2xX, since 1/2X and 1/(2X) are the same thing
it would be necessary to group 2X in parentheses
No, not necessary, since 2a=(2xa) by definition, alluded to in Cajori in 1928…
Sharp is a bit of an exception here, because all their other scientific calculators seem to
…follow all the rules of Maths, always. There’s something to be said for making sure you’re doing it right. :-)
Google uses the same priority for explicit and implicit multiplication
…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!)
Desmos and GeoGebra try to force the user into using fractions (which is a good design decision if you ask me)
It’s not, because a ÷ isn’t a fraction bar. They’re joining 2 terms into one and thus sometimes changing the answer
A lot of other tools like programming languages, spreadsheets, etc. don’t allow implicit multiplication syntax at all
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
let you choose if you want weak or strong juxtaposition
…which is a red flag to not use that calculator!
This gives you more control about how you like the calculator to behave in these situations
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
Wolfram|Alpha only uses strong juxtaposition between named variables, but weak juxtaposition for everything else. This might seem strange and inconsistent at first but is probably the least surprising behaviour for most people
I find any exceptions to following the rules of Maths surprising! No, you can’t just make up your own rules
many textbooks, “a/bc” is intended to denote a/(bc)
a/bc=a/(bc) in every textbook
Wolfram Language, it means (a/b)×c
Welcome to “we’re gonna add brackets to what you typed in and change the answer”
a multiplication sign has been omitted
…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
Multiplication and division have the same priority, they are “mathematically speaking” the same operation. This also applies to addition and subtraction. One is just the inverse function of the other
Yep, and The Distributive Law and Factorising are the inverse of each other
no rule about “multiplication before division” or “division before multiplication” they always have the same priority
…and Brackets is always first, so in this case it doesn’t even matter
In no way do any of the mnemonics represent any standard or norm in mathematics
Yes they do - mnemonics represent the actual order of operations rules
most children don’t become mathematicians later in life and if they do, they will learn all the other important stuff about the order of operations later
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
it’s hard to pump so much knowledge into children and teenagers
…and yet have you not noticed that teenagers almost never get this wrong - only adults do
Using “PEMDAS” to argue about the order of operations in mathematics
…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
Math notations and conventions evolve exactly like natural languages
No they don’t. Maths is universal
A lot of it is heavily based on historical thanks and work from previous generations
It’s all based on definitions and proofs, which are immutable
There is no definitive norm, standard or convention of notations and order of operations
You can find them in any high school textbook in your country (notation varies by country, but the rules don’t)
some words only appear in half of them (like “implicit multiplication by juxtaposition”)
“implicit multiplication” doesn’t appear in any Maths textbooks
sentences like “I saw the man with the telescope”, because it’s not clear if you saw him through the telescope or saw him holding (or looking through) a telescope
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
it should also be clear why there are no arguments or proofs for any side
But there are proofs! (There you go again with the “there is no…” red flag) Order of operations proof
