this post was submitted on 03 Dec 2023
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lol, math is literally the only subject that has rules set in stone. This example is specifically made to cause confusion. Division has the same priority as multiplication. You go from left to right. problem here is the fact that you see divison in fraction form way more commonly. A fraction could be writen up as (x)/(y) not x/y (assuming x and y are multiple steps). Plain and simple.
The fact that some calculator get it wrong means that the calculator is wrongly configured. The fact that some people argue that you do () first and then do what's outside it means that said people are dumb.
They managed to get me once too, by everyone spreading missinformation so confidently. Don't even trust me, look up the facts for yourself. And realise that your comment is just as incorrect as everyone who said the answer is 1. (uhm well they don't agree on 0^0, but that's kind of a paradox)
If we had 1/2x, would you interpret that as 0.5x, or 1/(2x)?
Because I can guarantee you almost any mathematician or physicist would assume the latter. But the argument you're making here is that it should be 0.5x.
It's called implicit multiplication or "multiplication indicated by juxtaposition", and it binds more tightly than explicit multiplication or division. The American Mathematical Society and American Physical Society both agree on this.
BIDMAS, or rather the idea that BIDMAS is the be-all end-all of order of operations, is what's known as a "lie-to-children". It's an oversimplification that's useful at a certain level of understanding, but becomes wrong as you get more advanced. It's like how your year 5 teacher might have said "you can't take the square root of a negative number".
An actual mathematician or physicist would probably ask you to clarify because they don't typically write division inline like that.
That said, Wolfram-Alpha interprets "1/2x" as 0.5x. But if you want to argue that Wolfram-Alpha's equation parser is wrong go ahead.
https://www.wolframalpha.com/input?i=1%2F2x
I will happily point out that Wolfram Alpha does this wrong. So do TI calculators, but not Casio or Sharp.
Go to any mathematics professor and give them a problem that includes 1/2x and ask them to solve it. Don't make it clear that merely asking "how do you parse 1/2x?" is your intent, because in all likelihood they'll just tell you it's ambiguous and be done with it. But if it's written as part of a problem and they don't notice your true intent, you can guarantee they will take it as 1/(2x).
Famed physicist Richard Feynman uses this convention in his work.
In fact, even around the time that BIDMAS was being standardised, the writing being done doing that standardisation would frequently use juxtaposition at a higher priority than division, without ever actually telling the reader that's what they were doing. It indicates that at the time, they perhaps thought it so obvious that juxtaposition should be performed first that it didn't even need to be explained (or didn't even occur to them that they could explain it).
According to Casio, they do juxtaposition first because that's what most teachers around the world want. There was a period where their calculators didn't do juxtaposition first, something they changed to because North American teachers were telling them they should, but the outcry front the rest of the world was enough for them to change it back. And regardless of what teachers are doing, even in America, professors of mathematics are doing juxtaposition first.
I think this problem may ultimately stem from the very strict rote learning approach used by the American education system, where developing a deeper understanding of what's going on seems to be discouraged in favour of memorising facts like "BIDMAS".
To be clear, I'm not saying 1/2x being 1/(2x) rather than 0.5x is wrong. But it's not right either. I'm just pretty firmly in the "inline formulae are ambiguous" camp. Whichever rule you pick, try to apply it consistently, but use some other notation or parenthesis when you want to be clearly understood.
The very fact that this conversation even happens is proof enough that the ambiguity exists. You can be prescriptive about which rules are the correct ones all you like, but that's not going to stop people from misunderstanding. If your goal is to communicate clearly, then you use a more explicit notation.
Even Wolfram Alpha makes a point of restating your input to show how it's being interpreted, and renders "1/2x" as something more like
to make very clear what it's doing.
This is definitely the best thing to do. It's what Casio calculators do, according to those videos I linked.
My main point is that even though there is theoretically an ambiguity there, the way it would be interpreted in the real world, by mathematicians working by hand (when presented in a way that people aren't specifically on the lookout for a "trick") would be overwhelmingly in favour of juxtaposition being evaluated before division. Maybe I'm wrong, but the examples given in those videos certainly seem to point towards the idea that people performing maths at a high level don't even think twice about it.
And while there is a theoretical ambiguity, I think any tool which is operating counter to how actual mathematicians would interpret a problem is doing the wrong thing. Sort of like a dictionary which decides to take an opinionated stance and say "people are using the word wrong, so we won't include that definition". Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
You're literally arguing that what you consider the ideal should be rigidly adhered to, though.
"How mathematicians do it is correct" is a fine enough sentiment, but conveniently ignores that mathematicians do, in fact, work at WolframAlpha, and many other places that likely do it "wrong".
The examples in the video showing inline formulae that use implicit priority have two things in common that make their usage unambiguous.
First, they all are either restating, or are derived from, formulae earlier in the page that are notated unambiguously, meaning that in context there is a single correct interpretation of any ambiguity.
Second, being a published paper it has to adhere to the style guide of whatever body its published under, and as pointed out in that video, the American Mathematical Society's style guide specifies implicit priority, making it unambiguous in any of their published works. The author's preference is irrelevant.
Also, if it's universally correct and there was no ambiguity in its use among mathematicians, why specify it in the style guide at all?
Mathematicians know wolfram is wrong and it was warned in my maths degree that you should "over bracket" in WA to make yourself understood. They tried hard to make it look like handwritten notation because reading maths from a word processor is typically tough and that creates the odd edge case like this.
1/2x does not equal 0.5x or it'd be written x/2 and I challenge you to find a mathematician who would argue differently. There's no ambiguity and claiming there is because anyone anywhere is having this debate is like claiming the world isn't definitely round because some people argue its flat.
Sometimes people are wrong.
Woo hoo! I hadn't heard of anyone else pointing this out (rather, I'm always on the receiving end of "But Wolfram says..."), so thanks for this comment! :-) Now I know I'm not alone in knowing that Wolfram is wrong.
OMG, I've run into so many people like that. They seem to believe (via saying "look, this blog says it's ambiguous too") that 2 wrongs make a right. No, you're both just wrong! Wolfram, Google, ChatGPT(!), the guy who should mind his own business, are all wrong.
Yes, they are... and unfortunately a whole bunch of the time they're unwilling to face it and/or admit it, even when faced with Maths textbooks which clearly show what they said is wrong.
Yes we do.
Wolfram-Alpha’s equation parser is wrong
Dude, this thread is four months old and I've gotten several notifications over the past week from you sporadically responding to comments I barely remember making. Find something better to do with your time than internet argument archeology. I'll even concede the point if it helps make you go away.
Thanks for the correction, you are right.
The point is the correction, not who made it. Almost every e-calculator is wrong, so people need to stop trusting them. They're no more accurate than GPT. Use a proper calculator.
I don't care.
go past past high school and this isn't remotely true
there are areas of study where 1+1=1
...given specific axioms. No rules are being broken.
but which axioms you decide are in use is an arbitrary choice
In modular arithmetic you can make 1+1=0 but I'm struggling to think of a situation where 1+1=1 without redefining the + and = functions.
Not saying you're wrong, but do you have an example? I'd be interested to see
But this is a high school Maths question, so "past high school" isn't relevant here.
Off topic, but the rules of math are not set in stone. We didn't start with ZFC, some people reject the C entirely, then there is intuitionistic logic which I used to laugh at until I learned about proof assistants and type theory. And then there are people who claim we should treat the natural numbers as a finite set, because things we can't compute don't matter anyways.
On topic: Parsing notation is not a math problem and if your notation is ambiguous or unclear to your audience try to fix it.
The rules around order of operations are!
Nothing ambiguous in this expression.
Indeed, it does.
No, it isn't. It simply tests who has remembered all the rules of Maths and who hasn't.
And there's no multiplication here - only brackets and division (and addition within the brackets).
Neither of those. A fraction could only be written inline as (x/y) - both of the things you wrote are 2 terms, not one. i.e. brackets needed to make them 1 term.
...they know all the relevant rules of Maths
You can find them here
and 1 is 100% correct.
Yes they do - it's 1 (it's the 5th index law). You might be thinking of 0/0, which depends on the context (you need to look at limits).
are you drunk?
Knowing the rules of Maths means someone might be drunk? Interesting conclusion.
Fuck it, I'm gonna waste time on a troll on the internet who's necroposting in te hopes that they actually wanna argue the learning way.
I said this because of the confusion around the division sign. Almost everyone at some point got it confused, or is just hell bent that one is corrent the other is not. In reality, it is such a common "mistake" that ppl started using it. I'm talking about the classic
4/2x. Ifx = 2, it is:4/2*2 = 2*2 = 44/(2*2) = 4/4 = 1Wolfram solved this with going with the second if it is an X or another variable as it's more intuitive.
Are you sure ur not a troll? how do you calculate
2(1+1)? It's4. It's called implicit multiplication and we do it all the time. It's the same logic that if a number doesn't have a sign it's positive. We could write this up as+2*(+1+(+1)), but it's harder to read, so we don't.I don't even fully understand you here. If we have a faction; at the top we have
1+2and at the bottom we have6-3. inline we could write this as(1+2)/(6-3). The result is 1 as if we simplify it's3/3.You can't say it's
((1+2)/(6-3)). It's the same thing. You will do the orders differently, but I can't think of a situation where it's incorrect, you are just making things harder on yourself.You fell into the 2nd trap too. If there is a letter or number or anything next to a bracket, it's multiplication. We just don't write it out, as why would we, to make it less readable?
2xis the same as2*xand that's the same as2(x).I can't even, you linked social media. The #1 most trust worthy website. Also I can't even read this shit. This guy talks in hashtags. I won't waste energy filtering out all the bullshit to know if they are right or wrong. Don't trust social media. Grab a calculator, look at wolfram docs, ask a professor or teacher. Don't even trust me!
I chose a side. But that side it the more RAW solution imo. let's walk it thru:
8/2(2+2), let's remove the confusion8/2*(2+2), brackets8/2*(4), mult & div, left -> right4*(4), let go4*4, the only16, answerBUT, and as I stated above IF it'd be like:
8/2xwithx=2+2then, we kinda decided to put implicit brackets there so it's more like8/(2x), but it's just harder to read, so we don't.And here is the controversy, we are playing the same game. Because there wasn't a an explicit multilication, you could argue that it should be handled like the scenario with the x. I disagree, you agree. But even this argument of "like the scenario with the x" is based of what Wolfram decided, there are no rules of this, you do what is more logical in this scenario. It can be a flaw in math, but it never comes up, as you use fractions instead of inline division. And when you are converting to inline, you don't spear the brackets.
You said it yourself, if we
lim (x->0) y/xthen there is an answer. But we aren't in limits.x/0in undefined at all circumstances (I should add that idk abstract algebra & non-linear geometry, idk what happens there. So I might be incorrect here).And by all means, correct me if I'm wrong. But link something that isn't an unreadable 3 parted mostodon post like it's some dumb twitter argument. This is some dumb other platform argument. Or don't link anything at all, just show me thru, and we know math rules (now a bit better) so it shouldn't be a problem... as long as we are civilised.
side note: if I did some typos.. it's 2am, sry.
It's the latter, as per the definition of Terms. There are references to this definition being used going back more than 100 years.
Yes, they do if it's 2x, but not if it's 2(2+2) - despite them mathematically being the same thing - leading to wrong answers to expressions such as the OP. In fact, that's true of every e-calculator I've ever seen, except for MathGPT (Desmos used to handle it correctly, but then they made a change to make it easier to enter fractions, and consequently broke evaluating divisions correctly).
No, it's not called implicit multiplication. It's distribution.
No, you can't. Adding that multiplication has broken it up into 2 terms. You either need to not add the multiply, or add another set of brackets if you do, to keep it as 1 term.
If a=2 and b=3, then...
1/axb=3/2
1/ab=(1/6)
No, it's distribution. Multiplication refers literally to multiplication signs, of which there aren't any in this expression.
No, 2A is the same as (2xA). i.e. it's a single Term. 2xA is 2 Terms (multiplied).
If a=2 and b=3, then...
axb=2x3 (2 terms)
ab=6 (1 term)
Only in the first post in each thread, so that people following those hashtags will see the first post, and can then click on it if they want to see the rest of the thread. Also "this guy" is me. :-)
I'm a Maths teacher with a calculator and many textbooks - I'm good. :-) Also, if you'd clicked on the thread you would've found textbook references, historical Maths documents, proofs, the works. :-)
2 mistakes here. Adding the multiplication sign in the 2nd step has broken up the term in the denominator, thus sending the (2+2) into the numerator, hence the wrong answer (and thus why we have a rule about Terms). Then you did division when there was still unsolved brackets left, thus violating order of operations rules.
But that's exactly what we do (but no extra brackets needed around 2x nor 2(2+2) - each is a single term).
Which is what the rules of Maths tells us to do - treat a single term as a single term. :-)
Yeah, there is. :-)
No, never. A fraction is a single term (grouped by a fraction bar) but division is 2 terms (separated by the division operator). Again it's the definition of Terms.
Have done, and appreciate the proper conversation (as opposed to those who call me names for simply pointing out the actual rules of Maths).
No problem. I t doesn't go into as much detail as the Mastodon thread though, but it's a shorter read (overall - with the Mastodon thread I can just link to specific parts though, which makes it handier to use for specific points), just covering the main issues.
Thanks, appreciated.
Idk where you teach, but I'm thankful you didn't teach me.
Let me quizz you, how do you solve
2(2+2)^2? because acording to your linked picture, because brackets are leftmost you do them first. If I were to believe you:(2*2+2*2)^2(4+4)^2, = 64but it's just simply incorrect.
2(4)^2, wow we're at a2x^22*16= 32The thing that pisses me off most, is the fact that, yes. Terms exists, yes they have all sorts of properties. But they are not rules, they are properties. And they only apply when we have unknows and we're at the most simplified form. For example your last link, the dude told us that those terms get prio because they are terms!? There are no mention of term prio in the book. It just simply said that when we have a simplified expression like:
2x^2+3x+5we call2x^2and3xand5terms. And yes they get priority, not because we named them those, but because they are multiplications. These help us at functions the most. Where we can assume that the highest power takes the sign at infinity. Maybe if the numbers look right, we can guess where it'd switch sign.I don't even want to waste energy proofreading this, or telling you the obvious that when we have a div. and a mult. and no x's there really is no point in using terms, as we just get a single number.
But again, I totally understand why someone would use this, it's easier. But it's not the rule still. That's why at some places this is the default. I forgot the name/keywords but if you read a calculator's manual there must be a chapter or something regarding this exact issue.
So yeah, use it. It's good. Especially if you teach physics. But please don't go around making up rules.
As for your sources, you still linked a blog post.
No, not because leftmost (did I say leftmost? No, I did not), because brackets. Brackets are always first in order of operations.
No, we're at x^2, because 2(4) is a bracketed term, and order of operations rules is brackets before exponents, and to solve the brackets we have to distribute the 2, so 2(4)^2=(2x4)^2=8^2=64.
Depends. The Distributive Property is a property, but The Distributive Law is a rule. Properties explain how/why things work, but rules have to be obeyed if you want to get the right answer. Terms is a rule, based on properties (similarly, The Distributive Law is a rule, which makes use of the Distributive Property).
Are you referring to pronumerals? Textbooks are quite explicit that the same rules apply to pronumerals as to numerals (since pronumerals literally stand-in for numerals).
Not priority, they are already fully solved because they are terms. If we have 2a, then there's literally nothing to be done (except substitute a value for a if you've been told what it is). 2xa on the other hand needs to be multiplied (2 terms separated by a multiplication).
Noted that you ignored where I pointed out why it makes a difference
Which book? I don't know what you're talking about now.
AKA Terms. And Terms are not expressions. Expressions are defined as being made up of Terms and operators. See previous textbook screenshot. 2a is a Term, 2xa is an Expression. And yes, you are right that a Term is a simplified expression, and being simplified, there is no further simplification to be done.
No, they are Terms. There is no multiplication. Multiplication refers literally to multiplication symbols. A Term is a product. i.e. the result of a multiplication. That's why they don't have multiplication symbols in them - it has already been done.
EXACTLY!! When a=2 and b=3, ab=6, a single number. AKA a Term.
We use it because that's how Maths works, and is a rule taught in all the textbooks, and has been for more than a century.
The name is Term. You can read about this exact issue in Maths textbooks.
I teach Maths, on which much of Physics is built.
In other words, you didn't even read it. The sources are in it - there are Maths textbooks in it.
Alright there buddy, I'd like to close this.
It's clear that your a troll. However, on the offchance that you didn't know, I'll tell you where you went wrong on the first one.
2(4)^2=(2x4)^2=8^2=64You can't distribute into a bracket, that's raised to the power of anything other than 1, like this. To do this you need to raise distributed number to the bracket's power's inverse. in this case 1/2.
2(4)^2=(2^(1/2)*4)^2=(sqrt(2)*4)^2=2*4^2=2*16=322*16=32Maybe if we look at it with roots you'd get it. wolfram syntax
2(4)^2=2Surd[4,1/2]2Surd[4,1/2]= Surd[4*2^(1/2),1/2]= (4*sqrt(2))^2= 4^2*2= 16*2= 32I hope you don't get scared from this math, you're a teacher afterall. I have no Idea how you could have gotten a degree or not kicked from school on day 1. Unless.. you are trolling me, fuck you for that. If you respond with more bullshiting, I'll block you.
Yes, that's right. Brackets before Exponents, as per the order of operations rules.
You know that's literally what The Distributive Law says you must do, right? Unless you have a source somewhere saying there's an exception?
Apparently you didn't bother reading any of the links I gave you, so here's one of the many textbooks which says you must distribute...
In case that's unclear, that means that 2x² and 2(x)² aren't the same thing (since 2(x)=(2x) by definition).
You know Wolfram disobeys The Distributive Law, right? I know I'm not the only one who knows this. Is that why you're insisting your way is right? Cos they're known to be wrong about this.
You call quoting Maths textbooks "bullshiting"?