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 :)
@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).
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
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.
Just write it better.
6/(2(1+2))
If you really wanted extra brackets it’d be 6/(2)(1+2). Of course, since there’s only 1 term in the first brackets they’re redundant, hence 6/2(1+2) is the fully simplified form, and is the way it’s written in Maths textbooks.
I always hate any viral math post for the simple reason that it gives me PTSD flashbacks to my Real Analysis classes.
The blog post is fine, but could definitely be condensed quite a bit across the board and still effectively make the same points would be my only critique.
At it core Mathematics is the language and practices used in order to communicate numbers to one another and it’s always nice to have someone reasonably argue that any ambiguity of communication means that you’re not communicating effectively.
The blog post is fine
Except that it’s wrong. Read this instead.
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!
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.
2(1+2) is the same as (1+2)+(1+2)
You nearly had it. 2(1+2) is the same as (2x1+2x2). The Distributive Law - it’s the reverse process to factorising.
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.
Division comes before Multiplication, doesn’t it? I know BODMAS.
This actually explains alot. Murica is Pemdas but Canadian used Bodmas so multiply is first in America.
As far as I understand it, they’re given equal weight in the order of operations, it’s just whichever you hit first left to right.
Ah, but if you use the rules BODMSA (or PEDMSA) then you can follow the letter order strictly, ignoring the equal precedence left-to-right rule, and you still get the correct answer. Therefore clearly we should start teaching BODMSA in primary schools. Or perhaps BFEDMSA. (Brackets, named Functions, Exponentiation, Division, Multiplication, Subtraction, Addition). I’m sure that would remove all confusion and stop all arguments. … Or perhaps we need another letter to clarify whether implicit multiplication with a coefficient and no symbol is different to explicit multiplication… BFEIDMSA or BFEDIMSA. Shall we vote on it?
Don’t need any extra letters - just need people to remember the rules around expanding brackets in the first place.
Obviously more letters would make the mnemonic worse, not better. I was making a joke.
As for the brackets ‘the rules around expanding brackets’ are only meaningful in the assumed context of our order of operations. For example, if we instead all agreed that addition should be before multiplication, then a×(b+c) would “expand” to a×b+c, because the addition is before multiplication anyway and the brackets do nothing.
I was making a joke.
Fair enough, but my point still stands.
if we instead all agreed that addition should be before multiplication
…then you would STILL have to do multiplication first. You can’t change Maths by simply agreeing to change it - that’s like saying if we all agree that the Earth is flat then the Earth is flat. Similarly we can’t agree that 1+1=3 now. Maths is used to model the real world - you can’t “agree” to change physics. You can’t add 1 thing to 1 other thing and have 3 things now, no matter how much you might want to “agree” that there is 3, there’s only 2 things. Multiplying is a binary operation, and addition is unary, and you have to do binary operators before unary operators - that is a fact that no amount of “agreeing” can change. 2x3 is actually a contracted form of 2+2+2, which is why it has to be done before addition - you’re in fact exposing the hidden additions before you do the additions.
the brackets do nothing
The brackets, by definition, say what to do first. Regardless of any other order of operations rules, you always do brackets first - that is in fact their sole job. They indicate any exceptions to the rules that would apply otherwise. They perform no other function. If you’re going to no longer do brackets first then you would simply not use them at all anymore. And in fact we don’t - when there are redundant brackets, like in (2)(1+2), we simply leave them out, leaving 2(1+2).
I think anything after (whichever grade your country introduces fractions in) should exclusively use fractions or multiplication with fractions to express division in order to disambiguate. A division symbol should never be used after fractions are introduced.
This way, it doesn’t really matter which juxtaposition you prefer, because it will never be ambiguous.
Anything before (whichever grade introduces fractions) should simply overuse brackets.
This comment was written in a couple of seconds, so if I missed something obvious, feel free to obliterate me.
A division symbol should never be used after fractions are introduced.
But a fraction is a single term, 2 numbers separated by a division is 2 terms. Terms are separated by operators and joined by grouping symbols.
I think weak juxtaposition is more easily taught
Except it breaks the rules which already are taught.
the PEMDAS ruleset
But they’re not rules - it’s a mnemonic to help you remember the actual order of operations rules.
Just let one die. Kill it, if you have to
Juxtaposition - in either case - isn’t a rule to begin with (the 2 appropriate rules here are The Distributive Law and Terms), yet it refuses to die because of incorrect posts like this one (which fails to quote any Maths textbooks at all, which is because it’s not in any textbooks, which is because it’s wrong).
Except it breaks the rules which already are taught.
It isn’t, because the ‘currently taught rules’ are on a case-by-case basis and each teacher defines this area themselves. Strong juxtaposition isn’t already taught, and neither is weak juxtaposition. That’s the whole point of the argument.
But they’re not rules - it’s a mnemonic to help you remember the actual order of operations rules.
See this part of my comment: “To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).”
Juxtaposition - in either case - isn’t a rule to begin with (the 2 appropriate rules here are The Distributive Law and Terms), yet it refuses to die because of incorrect posts like this one (which fails to quote any Maths textbooks at all, which is because it’s not in any textbooks, which is because it’s wrong).
You’re claiming the post is wrong and saying it doesn’t have any textbook citation (which is erroneous in and of itself because textbooks are not the only valid source) but you yourself don’t put down a citation for your own claim so… citation needed.
In addition, this issue isn’t a mathematical one, but a grammatical one. It’s about how we write math, not how math is (and thus the rules you’re referring to such as the Distributive Law don’t apply, as they are mathematical rules and remain constant regardless of how we write math).
It isn’t, because the ‘currently taught rules’ are on a case-by-case basis and each teacher defines this area themselves
Nope. Teachers can decide how they teach. They cannot decide what they teach. The have to teach whatever is in the curriculum for their region.
Strong juxtaposition isn’t already taught, and neither is weak juxtaposition
That’s because neither of those is a rule of Maths. The Distributive Law and Terms are, and they are already taught (they are both forms of what you call “strong juxtaposition”, but note that they are 2 different rules, so you can’t cover them both with a single rule like “strong juxtaposition”. That’s where the people who say “implicit multiplication” are going astray - trying to cover 2 rules with one).
See this part of my comment… Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler)
Yep, saw it, and weak juxtaposition would break the existing rules of Maths, such as The Distributive Law and Terms. (Re)learn the existing rules, that is the point of the argument.
citation needed
Well that part’s easy - I guess you missed the other links I posted. Order of operations thread index Text book references, proofs, the works.
this issue isn’t a mathematical one, but a grammatical one
Maths isn’t a language. It’s a group of notation and rules. It has syntax, not grammar. The equation in question has used all the correct notation, and so when solving it you have to follow all the relevant rules.
P.S. if you DID want to indicate “weak juxtaposition”, then you just put a multiplication symbol, and then yes it would be done as “M” in BEDMAS, because it’s no longer the coefficient of a bracketed term (to be solved as part of “B”), but a separate term.
6/2(1+2)=6/(2+4)=6/6=1
6/2x(1+2)=6/2x3=3x3=9
I don’t see the problem actually.
- Everything between ()
- Exponents
- multiply and devision
- plus and minus
- Always work from left to right.
==========
-
1+2= 3
-
No exponents
-
- 6 devised by 2 (whether a fraction or not) is 3
- 3 times 3 is 9
-
Nothing remains
The meme refers to the problem of handling implicit multiplication by juxtaposition.
Depending on what field you’re in, implicit multiplication takes priority over explicit multiplication/division (known as strong juxtaposition) rather than what you and a lot of people would assume (known as weak juxtaposition).With weak juxtaposition you end up 9 just as you did, but with strong juxtaposition you end up with 1 instead.
For most people and most scenarios this doesn’t matter, as you’d never encounter such ambiguous equations outside of viral puzzles like this, but it is worth knowing that not all fields agree on how implicit multiplication is handled.
The meme refers to the problem of handling implicit multiplication
There’s no such thing as implicit multiplication. dotnet.social/@SmartmanApps/110925761375035558
I don’t see the problem actually.
Everything between ()
You recreated the problem right there - ignored The Distributive Law. a(b+c)=(ab+ac). i.e. 2(1+2)=(2x1+2x2). After step 1 - solving brackets - all that’s left is 6/6. dotnet.social/@SmartmanApps/110819283738912144
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.
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.
If you’d ever taken any advanced math, you’d see that the answer is 1 all day. The implicit multiplication is done before the division because anyone taking advanced math would see 2(1+2) as a term that must be resolved first. The answer still lies in the ambiguity of the way the problem is written though. If the author used fractions instead of that stupid division symbol, there would be no ambiguity. It’s either 6/2 x 3 = 9 or [6/(2x3)] = 1. Comment formatting aside, if someone put 6 in the numerator, and then did or did NOT put all the rest in the denominator underneath a horizontal bar, it would be obvious.
TL;DR It’s still a formatting issue, but 9 is definitely not the clear and only answer.
The answer still lies in the ambiguity of the way the problem is written though
But it’s not ambiguous, as per the reason you already gave.
If the author used fractions instead of that stupid division symbol
If you use fractions then the whole thing is a single term, if you use division it’s 2 terms.
9 is definitely not the clear and only answer
1 is definitely the only answer.
"The obelus is treated differently,” Church said. "It could mean ratios, division or numerator and denominator, and these all tweak the meaning of the symbol.”
This is the only symbols I’ve ever seen used (but feel free to provide a reference if you know of any where it isn’t - the article hasn’t provided any references)…
Ratio is only ever colon.
Division is obelus (textbooks/computers) or slash (computers, though if it’s text you can use a Unicode obelus).
Fraction is fraction bar (textbooks) or obelus/slash inside brackets (computers). i.e. (a/b).
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.
there’s a mutual agreement that it’s only approximately correct.
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.
The distributive law has nothing to do with brackets.
The distributive law can be written in PEMDAS as a(b+c) = ab + ac, or PEASMD as ab+c = (ab)+(ac). It has no relation to the notation in which it is expressed, and brackets are purely notational.
The distributive law has nothing to do with brackets
BWAHAHAHA! Ok then, what EXACTLY does it relate to, if not brackets? Note that I’m talking about The Distributive LAW - which is about expanding brackets - not the Distributive PROPERTY.
a(b+c) = ab + ac
a(b+c)=(ab+ac) actually - that’s one of the common mistakes that people are making. You can’t remove brackets unless there’s only 1 term left inside, and ab+ac is 2 terms.
ab+c = (ab)+(ac)
No, never. ab+c is 2 terms with no further simplification possible. From there all that’s left is addition (once you know what ab and c are equal to).
brackets are purely notational
Yep, they’re a grouping symbol. Terms are separated by operators and joined by grouping symbols.
it shows how there is no consensus
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.
There really is no agreed upon standard even amongst experts
Yes there is. It’s taught in literally every Year 7-8 Maths textbook (but apparently Texas Instruments don’t care about that).
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
There has apparently been historical disagreement over whether 6÷2(3) is equivalent to 6÷2x3
No, there hasn’t - that’s a false claim by a Youtuber (and others who repeated it) - it is equal to 6÷(2x3) as per The Distributive Law and Terms, and even as per the letter he quoted! Here is where I debunked that claim.
Are you referring to Presh Talwalkar or someone else? How about his reference for historical use, Elizabeth Brown Davis? He also references a Slate article by Tara Haelle. I’ve heard Presh respond to people in the past over questions like this, and I’d love to hear his take on such a debunking. I have a lot of respect for him.
Your “debunk” link seems to debunk a clear rule-change in 1917. I wouldn’t disagree with that. I’ve never heard the variant where there was a clear change in 1917. Instead, it seems there was historical vagueness until the rules we now accept were slowly consolidated. Which actually makes sense.
The Distributive Law obviously applies, but I’m seeing references that would still assert that (6÷2) could at one time have been the portion multiplied with the (3).
And again, from logic I come from a place of avoiding ambiguity. When there is a controversiallly ambiguous form and an undeniablely unambiguous form, the undeniably unambiguous form is preferable.
Yes, the guy who should mind his own business.
How about his reference for historical use
Are you talking about his reference to Lennes’ letter? Lennes’ letter actually completely contradicts his claim that it ever meant anything different.
Elizabeth Brown Davis
Haven’t seen that one. Do you have a link?
He also references a Slate article by
…a journalist. The article ALSO ignores The Distributive Law and Terms.
I wouldn’t disagree with that.
Thank you. And also thank you for being the first person to engage in a proper conversation about it here.
I’ve heard Presh respond to people in the past over questions like this
I’ve seen him respond to people who agree with him. People who tell him he’s wrong he usually ignores. When he DOES respond to them he simply says “The Distributive Property doesn’t apply”. We’re talking about The Distributive LAW, NOT the Distributive Property. It’s called “law” for a reason. i.e. ALWAYS applies. I’ve only ever seen him completely unwilling to engage in any conversation with anyone who points out he’s wrong (contradicting his claim that he “welcomes debate”).
I have a lot of respect for him
Really?? Why’s that? I’m genuinely curious.
I’ve never heard the variant where there was a clear change in 1917
Me either. As far as I can tell it’s just people parroting his misinterpretation of Lennes’ letter.
Instead, it seems there was historical vagueness until the rules we now accept were slowly consolidated
I can’t agree with that. Lennes’ letter shows the same rules in 1917 as we use now. Cajori says the order of operations rules are at least 400 years old, and I have no reason to suspect they changed at all during that time period either. They’re all a natural consequence of the way we have defined the symbols in the first place.
The Distributive Law obviously applies
Again, thank you.
I’m seeing references that would still assert that (6÷2) could at one time have been the portion multiplied with the (3)
If it was written (6÷2)(1+2), absolutely that is the correct thing to do (expanding brackets), but not if it’s written 6÷2(1+2). If you mean the latter then I’ve never seen that - links?
It’s also clearly not a bug as some people suggest. Bugs are – by definition – unintended behavior.
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.
No matter how many times I explain that this is a notation for multiplication
It ISN’T a notation for multiplication - it’s a notation for a factorised term, and if you ignore The Distributive Law going back the other way then you just broke the factorised term dotnet.social/@SmartmanApps/110886637077371439
any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity.
This one already does have no ambiguity.
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.
a/bc is equally as ambiguous as a/b*c
It’s not ambiguous at all. By the definition of Terms - ab=(axb) - a/bc is 2 terms and a/bxc is 3 terms. If we were to write it in fraction form (to illustrate the difference), in the former c is in the denominator, but in the latter it’s in the numerator, hence a different answer. dotnet.social/@SmartmanApps/110846452267056791
you seem to take the position that the operations are resolved from left to right… but I haven’t seen this defined in mathematics anywhere
It applies to operators, or more precisely division. When doing the divisions, you have to do them left-to-right, but other than that each of the operators can be done in any order. i.e. it doesn’t matter what order you do the multiplications in, as long as you do them before the additions and subtractions. Unfortunately I’ve seen many people misremember left-to-right as an overarching rule, rather than only applying to division.
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.
basic calculator to solve multi part problems
This isn’t a multi-part problem, and any basic calculator other than Texas Instruments gets it correct.
These things are almost always written as fractions
Fractions are always written as fractions - they are 1 term - 2 separate terms are always separated by an operator, such as a division sign, like in this case.
the Kahn Academy or something similar.
Good advice! In particular look up what they say about The Distributive Law.
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.
How are people upvoting you for refusing to read the article?
Because those people also didn’t read the article and are reacting from their gut.
are reacting from their gut
As was the person who wrote the article. Did you not notice the complete lack of Maths textbooks in it?
I did read the article. I am commenting that I have never encountered strong juxtaposition and sharing why I think it is a poor choice.
I have never encountered strong juxtaposition
There’s “strong juxtaposition” in both Terms and The Distributive Law - you’ve never encountered either of those?
You probably missed the part where the article talks about university level math, and that strong juxtaposition is common there.
I also think that many conventions are bad, but once they exist, their badness doesn’t make them stop being used and relied on by a lot of people.
I don’t have any skin in the game as I never ran into ambiguity. My university professors simply always used fractions, therefore completely getting rid of any possible ambiguity.
You probably missed the part where the article talks about university level math,
This is high school level Maths. It’s not taught at university.
Because as a high school Maths teacher as soon as I saw the assertion that it was ambiguous I knew the article was wrong. From there I scanned to see if there were any Maths textbooks at any point, and there wasn’t. Just another wrong article.
Lol. Read it.
Why would I read something that I know is wrong? #MathsIsNeverAmbiguous
Mathematical notation however can be. Because it’s conventions as long as it’s not defined on the same page.
Mathematical notation however can be.
Nope. Different regions use different symbols, but within those regions everyone knows what each symbol is, and none of those symbols are in this question anyway.
Because it’s conventions as long as it’s not defined on the same page
The rules can be found in any high school Maths textbook.
Let’s do a little plausibility analysis, shall we? First, we have humans, you know, famously unable to agree on an universal standard for anything. Then we have me, who has written a PhD thesis for which he has read quite some papers about math and computational biology. Then we have an article that talks about the topic at hand, but that you for some unscientific and completely ridiculous reason refuse to read.
Let me just tell you one last time: you’re wrong, you should know that it’s possible that you’re wrong, and not reading a thing because it could convince you is peak ignorance.
I’m done here, have a good one, and try not to ruin your students too hard.
as an engineer with half a PhD
As an engineer with a full PhD. I’d say we engineers aren’t that great with math problems like this. Thus any responsible engineer would write it in a way that cannot be misinterpreted. Because misinterpreted mathematics can kill people…
As an engineer with a full PhD. I’d say we engineers aren’t that great with math problems like this
Yay for a voice of reason! I’ve yet to see anyone who says they have a Ph.D. get this correct (I’m a high school Maths teacher/tutor - I actually teach this topic).
Yeah, but implicit multiplication without a sign is often treated with higher priority.
Sure. That doesn’t mean it’s right to do.
Please read the article, that’s exactly what it’s about. There is no right answer.
I read the article, and it explained the situation and the resultant confusion very well. That said, could we not have some international body just make a decision one way or the other, instead of perpetuating this uncertainty?
could we not have some international body just make a decision one way or the other
There’s no decision to be made. The correct rules are already taught in literally every Year 7-8 Maths textbook.
There is a right answer. Read this instead dotnet.social/@SmartmanApps/110897908266416158
as a half PhD
Go read the article, it’s about you
Go read the article, it’s about you
The article is wrong dotnet.social/@SmartmanApps/110897908266416158
Without parentheses around (2×3)
But there is parentheses around (2x3). a(b+c)=(ab+ac) - The Distributive Law. You can’t remove them unless there is only 1 term left inside. You removed them when you still had 2 terms inside, 2x3.
6/2(1+2)=6/2(3)=6/(2*3)=6/6=1
OR
6/2(1+2)=6/(2+4)=6/6=1
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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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?
Here is an alternative Piped link(s):
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I’m open-source; check me out at GitHub.
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.”
Pretty much. While it’s worth knowing that not everyone agrees on how implicit multiplication is prioritised, anywhere that everyone agreeing on the answer actually mattered, you wouldn’t write an equation as ambiguous as this one in the first place
It’s not ambiguous. People who say it is have usually forgotten The Distributive Law or Terms, or more commonly both!
Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma.
Not even remotely similar. Maths rules are fixed. The order of operations rules are at least 400 years old.
mathematical notation is just as flexible as any other facet of written human communication
No, it isn’t. The book “A history of mathematical notation” is in itself more than 100 years old.
Wow neat, and yet the thread was full of people going back and forth about how the equation can be misinterpreted based on how the order of operations can be interpreted. Thanks for your months later input though.
I only just found the thread yesterday. There’s only 1 “interpretation”, and the only back and forth I’ve seen about interpretations is about implicit multiplication, which isn’t a thing, at all - it’s people conflating The Distributive Law and Terms dotnet.social/@SmartmanApps/110925761375035558
So you are saying exactly what I said; people can misinterpret things that other people have written. Good job. Thanks again for stopping by a 3 month old thread about a dumb meme.
So you are saying exactly what I said; people can misinterpret things that other people have written
No, I’m not. They’re “misinterpreting” something that isn’t even a rule of Maths. There’s no way to misinterpret the actual rules, there’s no way to misinterpret the equation. There’s no alternative interpretations of the notation. Someone who didn’t remember the rules literally made up “implicit multiplication”, and then other people argued with them about what that meant. 😂
You look like a real idiot here. I really suggest you actually read the article instead of “scanning” it. You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up. Implicit multiplication is not the controversial part of this equation, which you would know if you read the article and understood what people in this thread are even talking about. Stop spamming your shitty blog and just. Read. The. Article.
read the article instead of “scanning” it.
I stopped reading as soon as I saw the claim that the right answer was wrong. I then scanned it for any textbook references, and there were none (as expected).
You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up
Funny that you use the word “term”, since Terms are ONE of the things that people are referring to when they say “implicit multiplication” - the other being The Distributive Law. i.e. Two DIFFERENT actual rules of Maths have been lumped in together in a made-up rule (by people who don’t remember the actual rules).
BTW if you think it’s not made-up then provide me with a Maths textbook reference that uses it. Spoiler alert: you won’t find any.
Implicit multiplication is not the controversial part of this equation
It’s not the ONLY controversial part of the equation - people make other mistakes with it too - but it’s the biggest part. It’s the mistake that most people have made.
shitty blog
So that’s what you think of people who educate with actual Maths textbook references?
Read. The. Article.
Read Maths textbooks.
