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 :)
FACT CHECK 4/5
a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line
There’s absolutely nothing wrong with doing that. The order of operations rules have everything covered. Anything which follows an operator is a separate term. Anything which has a fraction bar or brackets is a single term
most typical programming languages don’t allow omitting the multiplication operator
Because they don’t come with order of operations built-in - the programmer has to implement it (which is why so many e-calculators are wrong)
“.NET IDE0048 – Add parentheses for clarity”
Microsoft has 3 different software packages which get order of operations wrong in 3 different ways, so I wouldn’t be using them as an example! There are multiple rules of Maths they don’t obey (like always rounding up 0.5)
Let’s say we want to clean up and simplify the following statement … o×s×c×(α+β) … by removing the explicit multiplication sign and order the factors alphabetically: cos(α+β) Nobody in their right mind would remove the explicit multiplication sign in this case
This is wrong in so many ways!
- you did multiplication before brackets, which violates order of operations rules! You didn’t give enough information to solve the brackets - i.e. you left it ambiguous - you can’t just go “oh well, I’ll just do multiplication then”. No, if you can’t solve Brackets then you can’t solve ANYTHING - that is the whole point of the order of oeprations rules. You MUST do brackets FIRST.
- the term (α+β) doesn’t have a coefficient, so you can’t just randomly decide to give it one. It is a separate term from the rest Is there supposed to be more to this question? Have you made this deliberately ambiguous for example?
- if the question is just to simplify, then no simplification is possible. You’ve not given any values to substitute for the pronumerals
- (α+β) is presumably (you’ve left this ambiguous by not defining them) a couple of angles, and if so, why isn’t the brackets preceded by a trig function?
- As it’s written, it just looks like a straight-forward multiplying and adding pronumerals except you didn’t give us any values for the pronumerals meaning no simplfication is possible
- if this was meant to be a trig question (again, you’ve left out any information that would indicate this, making it ambiguous) then you wouldn’t use c, o, or s for your pronumerals - you’ve got a whole alphabet left you can use. Appropriate choice of pronumerals is something we teach in Maths. e.g. C for cats, D for dogs. You haven’t defined what ANY of these pronumerals are, leaving it ambiguous
Nobody will interpret cos(α+β) as a multiplication of four factors
- as originally written it’s 4 terms, not 1 term. i.e. it’s not cos(α+β), it’s actually oxsxxx(α+β), since that can’t be simplified. And yes, that’s 4 terms multiplied!
From those 7 points, we can see this is not a real Maths problem. You deliberately made it ambiguous (didn’t say what any of the pronumerals are) so you could say “Look! Maths is ambiguous!”. In other words, this is a strawman. If you really think Maths is ambiguous, then why didn’t you use a real Maths example to show that? Spoiler alert: #MathsIsNeverAmbiguous hence why you don’t have a real example to illustrate ambiguity
Implicit multiplications of variables with expressions in parentheses can easily be misinterpreted as functions
No they can’t. See previous points. If there is a function, then you have to define what it is. e.g. f(x)=x². If no function has been defined, then f is the pronumeral f of the factorised term f(x), not a function. And also, if there was a function defined, you wouldn’t use f as a pronumeral as well! You have the whole rest of the alphabet left to use. See my point about we teach appropriate choice of pronumerals
So, ambiguity really hides everywhere
No, it really doesn’t. You just literally made up some examples which go against the rules of Maths then claimed “Look! Maths is ambiguous!”. No, it isn’t - the rules of Maths make sure it’s never ambiguous
IMHO it would be smarter to only allow the calculation if the input is unambiguous.
Which is exactly what calculators do! If you type in something invalid (say you were missing a bracket), it would say “syntax error” or something similar
force the user to write explicit multiplications
Are you saying they shouldn’t be allowed to enter factorised terms? If so, why?
force notation that is never ambiguous
We already do
but that would lead to a very convoluted mess that’s hard to read and write
In what way is 6/2(1+2) either convoluted or hard to read? It’s a term divided by a factorised term - simple
providing context that makes it unambiguous
In other words, follow the rules of Maths.
Links about various potentially ambiguous math notations
Spoiler alert: they’re not
“Most ambiguous phrases and notations in maths”
e.g. fx=f(x), which I already addressed. It’s either been defined as a function or as pronumerals, therefore nothing ambiguous
“Absolute value notation is ambiguous”
No, it’s not. |a|b|c| is the absolute value of a, times b, times the absolute value of c… which you would just write as b|ac|. Unlike brackets you can’t have nested absolute values, so the absolute value of (a times the absolute value of b times c) would make no sense, especially since it’s the EXACT same answer as |abc| anyway!
In-line power towers like
Left associativity. i.e. an exponent is associated with the term to its left - solve exponents right to left
People saying “I don’t know how to interpret this” doesn’t mean it’s ambiguous, nor that it isn’t defined. It just means, you know, they need to look it up (or ask a Maths teacher)! If someone says “I don’t know what the word ‘cat’ means”, you don’t suddenly start running around saying “The word ‘cat’ is ambiguous! The word ‘cat’ is ambiguous!” - you just tell them to look it up in a dictionary. In the case of Maths, you look it up in a Maths textbook
Because the actual math is easy almost everybody has an opinion on it
…and any of them which contradict any of the rules of Maths are demonstrably wrong
Most people also don’t know that with weak and strong juxtaposition there are two conflicting conventions available
…and Maths teachers know that both of them are made-up and not real things in Maths
But those mnemonics cover just the basics. The actual real world is way more complicated and messier than “BODMAS”
Nope. The mnemonics plus left to right covers everything you need to know about it
Even people who know about implicit multiplication by juxtaposition dismiss a lot of details
…because it’s not a real thing
Probably because of confirmation bias and/or because they don’t want to invest so much time into thinking about stupid social media posts
…or because they’re a high school Maths teacher and know all the rules of Maths
the actual problem with the ambiguity can’t be explained in a quick comment
Yes it can…
Forgotten rules of Maths - The Distributive Law (e.g. a(b+c)=(ab+ac)) applies to all bracketed Terms, and Terms are separated by operators and joined by grouping symbols
Bam! Done! Explained in a quick comment
I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher
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 actually 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?
@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).
You state that the ambiguity comes from the implicit multiplication and not the use of the obelus.
I.e. That 6 ÷ 2 x 3 is not ambiguous
What is your source for your statement that there is an accepted convention for the priority of the iinline obelus or solidus symbol?
As far as I’m aware, every style guide states that a fraction bar (preferably) or parentheses should be used to resolve the ambiguity when there are additional operators to the right of a solidus, and that an obelus should never be used.
Which therefore would make it the division expressed with an obelus that creates the ambiguity, and not the implicit multiplication.
(Rest of the post is great)
In this case it’s actually the absence of sources. I couldn’t find a single credible source that states that ÷ has somehow a different operator priority than / or that :
The only things there are a lot of are social media comments claiming that without any source.
My guess is that this comes from a misunderstanding that the obelus sign is forbidden in a lot of standards. But that’s because it can be confused with other symbols and operations and not because the order of operations is somehow unclear.
What is your source for the priority of the / operator?
i.e. why do you say 6 / 2 * 3 is unambiguous?
Every source I’ve seen states that multiplication and division are equal priority operations. And one should clarify, either with a fraction bar (preferably) or parentheses if the order would make a difference.
Same priority operations are solved from left to right. There is not a single credible calculator that would evaluate “6 / 2 * 3” to anything else but 9.
But I challenge you to show me a calculator that says otherwise. In the blog are about 2 or 3 dozend calculators referenced by name all of them say the same thing. Instead of a calculator you can also name a single expert in the field who would say that 6 / 2 * 3 is anything but 9.
Will you accept wolfram alpha as credible source?
https://mathworld.wolfram.com/Solidus.html
Special care is needed when interpreting the meaning of a solidus in in-line math because of the notational ambiguity in expressions such as a/bc. Whereas in many textbooks, “a/bc” is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages such as the Wolfram Language, it means (a/b)×c. For clarity, parentheses should therefore always be used when delineating compound denominators.
The link references “a/bc” not “a/b*c”. The first is ambiguous, the second is not.
I just finished your article and wow! I’m definitely going to save it and share it the next time I come across another one of those viral problems. It was incredibly thorough and well researched, you clearly put a lot of energy and effort into it and it blew me away. It was really refreshing to see someone articulate themselves so passionately with supporting research. I look forward to reading more of your work!! 👏
That’s cool and Imma let you finish but I’m not a mathematician and the answer is 9.
That’s the correct answer if you follow one of the conventions. There are actually two conflicting but equally valid conventions. The blog explains the full story but this math problem is really ambiguous.
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’m not a mathematician and the answer is 9
I’m a Mathematician and the answer is 1.
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!
Nope it’s bedmas since everything is brackets
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)
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.
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.
The ambiguous ones at least have some discussion around it. The ones I’ve seen thenxouple times I had the misfortune of seeing them on Facebook were just straight up basic order of operations questions. They weren’t ambiguous, they were about a 4th grade math level, and all thenpeople from my high-school that complain that school never taught them anything were completely failing to get it.
I’m talking like 4+1x2 and a bunch of people were saying it was 10.
You guys are doing it all wrong: ask chatgpt for the correct answer and paste it here. Done.
Who needs to learn or know anything really?
ChatGPT’s Answer:
The expression 6/2(1+2) involves both multiplication and division. According to the order of operations (PEMDAS/BODMAS), you should perform operations inside parentheses first, then any multiplication or division from left to right.
Let’s break down the expression step by step:
Inside the parentheses: 1 + 2 = 3
Now the expression becomes 6/2 * 3
Division: 6/2 = 3
Multiplication: 3 * 3 = 9
So, 6/2(1+2) is equal to 9.
FACT CHECK 3/5
It’s only a matter of taste and how widespread a convention or notation is
The rules are in every high school Maths textbook. The notation for your country is in your country’s Maths textbooks
There are no arguments or proofs about what definition is correct
1+1=2 by definition (or whatever the notation is in your country). If you write 1+1=3 then that is wrong by definition
I found a lot of explanations online that were either half-assed or just plain wrong
And you seem to have included most of them so far - “implicit multiplication”, “weak juxtaposition”, “conventions”, etc.
You either were taught something wrong or you misremember it.
Spoiler alert: It’s always the latter
IMHO the mnemonics would be better without “division” and “subtraction”, because it would force people to think about it before blindly applying something the wrong way – “PEMA” for example. Parentheses, exponentiation, multiplication, addition
In fact what would happen is now people wouldn’t know in what order to do division and subtraction, having removed them from the mnemonic (and there’s absolutely no reason at all to remove them - you can do everything in the mnemonic order and it works, provided you also obey the left-to-right rule, which is there to make sure you obey left associativity)
parenthesis and exponents students typically don’t learn the order of operations through some mnemonics they remember them through exercise
That’s not true at all. Have you not read through some of these arguments? They’re all full of “Use BEDMAS!”, “Use PEMDAS!”, “It’s PEMDAS not BEDMAS!” - quite clearly these people DID learn order of operations through the mnemonics
trying to remember some random acronyms
There’s no requirement to memorise any acronym - you can always just make up your own if you find that easier! I did that a lot in university to remember things during the exam
they also state to “not use × to express a simple product”
…because a product is a Term, and to insert a x would break it into 2 Terms
A product is the result of a multiplication
The center dot also should not be used to mean a simple product
Exact same reason. They are saying “don’t turn 1 term into 2 terms”. To put that into the words that you keep using, “don’t use weak juxtaposition”
Nobody at the American Physical Society (at least I hope) would say that 6/2×3 equals one, because that’s just bonkers
Because it would break the rule of left associativity (i.e. left to right). No-one is advocating “multiplication before division” where it would violate left to right (usually by “multiplication” they’re actually referring to Terms, and yes, you literally always have to do Terms before Division)
÷ (obelus), : (colon) or / (solidus), but that is not the case and they can be used interchangeably without any difference in meaning. There are no widespread conventions, that would attribute different meanings
Yes there is. Some countries use : for divide, whereas other countries use it for ratio
most standards forbid multiple divisions with inline notation, for example expressions like this 12/6/2
Name one! Give me a reference! There’s nothing forbidding that in Maths (though we would more usually write it as 12/(6x2)). Again, all you have to do is obey left to right
Funny enough all the examples that N.J. Lennes list in his letter use
…Terms. Same as all textbooks do now
and thus his rule could be replaced by
…Terms, the already-existing rule that he apparently didn’t know about (he mentions them, and products, but manages to completely miss what that actually means)
“Something, something, distributive property, something ….”
Something, something, Distributive Law (yes, some people use the wrong name, but in talking about the property, not the law, you’re knocking down a strawman)
The distributive property is just a property that applies to some operations
…and The Distributive Law applies to every bracketed term that has a coefficient, in this case it’s 2(1+2)
It has nothing to do with the order of operations
And The Distributive Law has everything to do with order of operations, since solving Brackets is literally the first step!
I’ve no idea where this idea comes from
Maybe you should’ve asked someone. Hint: textbooks/teachers
because there aren’t any primary sources (at least I wasn’t able to find any)
Here it is again, textbook references, proofs, memes, the works
should be calculated (distributed) first
Bingo! Distribution isn’t Multiplication
6÷2(3). If we follow the strong juxtaposition convention, we must
…distribute the 2, always
It has nothing to do with the 3 being inside parentheses
It has everything to do with there being a coefficient to the brackets, the 2
Those parentheses are only there, because
…it’s a factorised term, and the opposite of factorising is The Distributive Law
the parentheses do not force the multiplication
No, it forces distribution of the coefficient. a(b+c)=(ab+ac)
The parentheses are only there to make it clear that
…it is a factorised term subject to The Distributive Law
we are implicitly multiplying two separate numbers.
They’re NOT 2 separate numbers. It’s a single, factorised term, in the same way that 2a is a single term, and in this case a is equal to (1+2)!
With the context that the engineer is trying to calculate the radius of a circle it’s clear that they meant r=C/(2π)
Because 2π is a single term, by definition (it’s the product of a multiplication), as is r itself, so that should actually be written r=(C/2π)
When symbols for quantities are combined in a product of two or more quantities, this combination is indicated in one of the following ways: ab,a b,a⋅b,a×b
Incorrect. Only the first one is a term/product (not separated by any operators) - the last 2 are multiplications, and the 2nd one is literally meaningless. Space isn’t defined as meaning anything in Maths
Division of one quantity by another is indicated in one of the following ways:
The first is a fraction
The second is a division
The third is also a fraction
The last is a multiplication by a fraction
Creates ambiguity since space isn’t defined to mean anything in Maths. Looks like a typo - was there meant to be a multiply where the space is? Or was there not meant to be a space??
By definition ab-1=a1b-1=(a/b)
