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 :)
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.
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.
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.
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!
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
FACT CHECK 1/X
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
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.
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.
Is it though? I’ve only ever seen it treated as standard multiplication.
Read TFA
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.
There is a right answer. Read this instead dotnet.social/@SmartmanApps/110897908266416158
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.
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
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
That’s cool and Imma let you finish but I’m not a mathematician and the answer is 9.
I’m not a mathematician and the answer is 9
I’m a Mathematician and the answer is 1.
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.
That’s the correct answer if you follow
…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
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.
I don’t agree with the notation of the American Physical Society
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:
the American Physical Society state in their Style and Notation Guide on page 21 that they do “multiplication before division”, but you must be careful to not take that out of context
What if the real answer is the friends we made along the way?
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.
Great read! Easy for everyone to understand, but also thorough. I loved the breakdown into the calculators functionality
Thank you for taking the time reading it.
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.
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?
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.
If you’d ever taken any advanced math, you’d see that the answer is 1 all day
Don’t need to do advanced Maths - every rule you need to know for this problem is taught in Year 7.
“Always remember to solve using PEMDAS once you’ve used the distributive property!” Link%20and%20subtraction%20(S).)
(emphasis mine)
And…? Not sure what your point is, but the link is VERY badly worded…
- The Distributive Law and The Distributive Property aren’t the same thing - he’s applying The Distributive Law, but mistakenly calling it The Distributive Property (a lot of people make that mistake). The latter is merely a property in Maths (like the commutative property, the associative property, etc.), the former an actual rule of Maths The Distributive Law
- Applying the Distributive Law - i.e. expanding brackets/parentheses - is part of solving brackets. i.e. the first step in BEDMAS/PEMDAS. There’s no “once you’ve used”, you’ve already started!
- As I already said, this is taught in Year 7, so I’m not sure what your point is?
That you’re still wrong? As I said, the true answer is that the problem is written poorly due to the obelus and thus is open to interpretation. You’re entitled to your own interpretation since it’s written poorly, I just find it pretty obviously less logical than multiplying using the distributive property first to resolve the term with the parentheses fully as you would in any advanced math.
Also, distributive law and distributive property are the same thing per Khan academy “The distributive property is sometimes called the distributive law of multiplication and division.”
Wait till you hear that “i before e except after c” wasn’t true either. It’s wild that you think 7th grade math overrules grad school math though lol.
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).
those calculators because that is a badly written equation
It’s not badly written, and the reason Texas Instruments gets it wrong is right there in their manual (disobeys The Distributive Law).
modern rules of math
The order of operations rules haven’t changed in at least 100 years, and more likely at least 400 years. Don’t listen to Youtubers who can’t cite a single Maths textbook.
“2(3)” is the same as “2 x 3”
No, it’s the same as (2x3), as per The Distributive Law and Terms.
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).
leaving us 3(3)
You just did division before brackets, which violates order of operations rules. 6÷2(3)=6÷(2x3)=6÷6=1
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?
While I agree the problem as written is ambiguous and should be written with explicit operators, I have 1 argument to make. In pretty much every other field if we have a question the answer pretty much always ends up being something along the lines of “well the experts do this” or “this professor at this prestigious university says this”, or “the scientific community says”. The fact that this article even states that academic circles and “scientific” calculators use strong juxtaposition, while basic education and basic calculators use weak juxtaposition is interesting. Why do we treat math differently than pretty much every other field? Shouldn’t strong juxtaposition be the precedent and the norm then just how the scientific community sets precedents for literally every other field? We should start saying weak juxtaposition is wrong and just settle on one.
This has been my devil’s advocate argument.
While I agree the problem as written is ambiguous
It’s not.
the answer pretty much always ends up being something along the lines of “well the experts do this” or “this professor at this prestigious university says this”, or “the scientific community says”.
Agree completely! Notice how they ALWAYS leave out high school Maths teachers and textbooks? You know, the ones who actually TEACH this topic. Always people OTHER THAN the people/books who teach this topic (and so always end up with the wrong conclusion).
while basic education and basic calculators use weak juxtaposition
Literally no-one in education uses so-called “weak juxtaposition” - there’s no such thing. There’s The Distributive Law and Terms, both of which use so-called “strong juxtaposition”. Most calculators do too.
Shouldn’t strong juxtaposition be the precedent and the norm
It is. In fact it’s the rules (The Distributive Law and Terms).
We should start saying weak juxtaposition is wrong
Maths teachers already DO say it’s wrong.
This has been my devil’s advocate argument.
No, this is mostly a Maths teacher argument. You started off weak (saying its ambiguous), but then after that almost everything you said is actually correct - maybe you should be a Maths teacher. :-)
I tried to be careful to not suggest that scientist only use strong juxtaposition. They use both but are typically very careful to not write ambiguous stuff and practically never write implicit multiplications between numbers because they just simplify it.
At this point it’s probably to late to really fix it and the only viable option is to be aware why and how this ambiguous and not write it that way.
As stated in the “even more ambiguous math notations” it’s far from the only ambiguous situation and it’s practically impossible (and not really necessary) to fix.
Scientist and engineers also know the issue and navigate around it. It’s really a non-issue for experts and the problem is only how and what the general population is taught.
Damn ragebait posts, it’s always the same recycled operation. They could at least spice it up, like the discussion about absolute value. What’s |a|b|c|?
What I gather from this, is that Geogebra is superior for not allowing ambiguous notation to be parsed 👌
Your example with the absolute values is actually linked in the “Even more ambiguous math notations” section.
Geogebra has indeed found a good solution but it only works if you input field supports fractions and a lot of calculators (even CAS like WolframAlpha) don’t support that.
Even more ambiguous math notations
Except that isn’t ambiguous either. See my reply to the original comment.
Geogebra has indeed found a good solution
Geogebra has done the same thing as Desmos, which is wrong. Desmos USED TO give correct answers, but then they changed it to automatically interpret / as a fraction, which is good, except when they did that it ALSO now interprets ÷ as a fraction, which is wrong. ½ is 1 term, 1÷2 is 2 terms (but Desmos now treats it as 1 term, which goes against the definition of terms)
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.
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.
if it was 6÷2x(2+1) they suggested do division and mult from left to right, but 6÷2(2+1)
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.
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.
Yeah, that’s why fractions are good thing.
Don’t forget math with fruits! https://imgur.com/JOuRhQ3
