this post was submitted on 03 Dec 2023
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this comment section illustrates perfectly why i hate maths so much lmao
love ambiguous, confusing rules nobody can even agree on!
The problem isn't math, it's the people that suck at at it who write ambigous terms like this, and all the people in the comments who weren't educated properly on what conventions are.
Yeah, you could easily make this more straightforward by putting parentheses around 8÷2. It's like saying literature sucks because Finnegans Wake is incomprehensible.
But that would be a different expression with a different answer (16 rather than 1). This is the mistake made by the programmer of the e-calc - treats it as though there's extra brackets there when there isn't.
Huge shout out to the jaded AF high school math teachers that don't give a fuck any more!
They do care. The issue is everyone argues about it without even asking Maths teachers about it to being with! I guarantee (I've seen it myself) literally every blog you read which says this is "ambiguous", without exception they never mention Maths textbooks or Maths teachers (because then they wouldn't be able to bombastically declare "This is ambiguous!").
It's not ambiguous
Everyone was taught the rules of Maths - it's just a matter of who remembers them or not.
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.
Here is an alternative Piped link(s):
Famed physicist Richard Feynman uses this convention in his work.
the writing being done doing that standardisation would frequently use juxtaposition at a higher priority than division
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I'm open-source; check me out at GitHub.
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.
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).
This is more language/writing style than math. The math is consistent, what’s inconsistent is there are different ways to express math, some of which, quite frankly, are just worse at communicating the mathematical expression clearly than others.
Personally, since doing college math classes, I don’t think I’d ever willingly write an expression like that exactly because it causes confusion. Not the biggest issue for a simple problem, much bigger issue if you’re solving something bigger and need combine a lot of expressions. Just use parentheses and implicit multiplication and division. It’s a lot clearer and easier to work with.
There's no such thing as implicit multiplication
Something about the way this thread was written was kind of confusing, so I don’t really get what their point was. Is it just that the terminology is wrong? Or am I missing something?
Like, whatever you call it, a x b, a*b, ab, and a(b) are all acceptable notations to describe the operation “multiply a and b.” Some are nicer to use than others depending on the situation.
Ok, sorry about that. I'm more than happy to update it if you want to give me some constructive feedback on what was confusing about it. Note though that this was the 3rd part in the series, and maybe you didn't go back and read the previous 2 parts? They start here
Yes and yes. :-) The 2 actual rules of Maths that apply here are Terms and The Distributive Law. These are 2 different rules of Maths - neither of which is "multiplication" - and so when lumping them together as "implicit multiplication" you end up with unpredictable, and usually wrong, answers. The only way to always get the right answer is to follow the actual rules of Maths.
No, they're not. The first two are multiplications, the second two are Terms. Note that a Term is a product, the result of a multiplication. In the mnemonics, "Multiplication"" refers literally to multiplication signs, and nothing else.
NP. I'm not really great at giving writing advice, so can't really help there. Something about it just didn't click when I read it. The extra context you linked did help a bit.
As far as the issue: After reading it I think it does just seem to be a matter of terminology mixed with problems that arise with when you need to write math expressions inline in text. If you can write things out on paper or use a markdown language, it's really easy to see how a fractional expression is structured.
8
2(1+3)
is a lot easier to read than 8/2(1+3) even if they technically are meant to be evaluated the same. There's no room for confusion.
And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they're just the same thing in practice. When I see a(x+1) I know that in order to multiply these I need to distribute. And if we fill in the algebraic symbols for numbers, you don't even need to distribute to get the answer since you can just evaluate the parentheses then use the result to multiply by the outside.
Conversely, if I was factoring something, I would need to do division.
ax + a
a
= x+1, thus: a(x+1)
I think we're basically talking about the same thing, I'm just being a bit lose with the terminology.
And while we're at it, the best way to make sure there's no misunderstanding is to just use parenthesis for EVERYTHING! I'm mostly kidding, this can rapidly get unreadable once you nest more than a few parens, although for these toy expressions, it would have the desired effect.
(8)/(2(1+3)) is obviously different than (8/2)(1+3)
But 8/2(1+3) isn't a fraction. The / - the computing equivalent of ÷ (which can only be written using Unicode on a computer, so a bit of a pain to use compared to / )- is an operator, which means they're 2 separate terms. A fraction bar is a grouping symbol, which means it's 1 term. In this particular case it doesn't matter, but if it appeared in a bigger expression then it absolutely does matter. The way to write 8/2(1+3) as a fraction inline is to add extra brackets. i.e. (8/2(1+3)) - because brackets are also a grouping symbol.
Bu they're not, for the same reason. Firstly, the Distributive Law isn't multiplication at all - which only applies literally to multiplication symbols - it applies to bracketed terms (i.e. is a single term which needs to be distributed) - and secondly it applies to a single term, whereas multiplication applies to 2 terms (one before and one after). Anyone who talks about 2(1+3) needing to be "multiplied" has already made the mistake that is going to lead to a wrong answer (unless they just happen to "multiply" before they divide, which is an accidental way to get the right answer).
Indeed, that is the precise reason the Distributive Law exists - they are the opposite operation to each other! Anyone who adds a multiplication symbol has broken up the factorised term, again leading to the wrong answer.
Yeah, and that's all I was pointing out in the first place - please don't use "implicit multiplication". The term itself - i.e. it includes "multiplication" - leads people to do it wrong (because they treat it as multiplication, not brackets, then argue about the precedence of "multiplication"!). It needs to die!
Well that's why the rules of Terms ab=(axb) and The Distributive Law a(b+c)=(a*(b+c))=(ab+ac) exist to begin with - less brackets! :-) Imagine having to write a fraction as (1/(axb)) all the time!
Correct, though a lot of people treat it as the latter (yet another way to do it wrong - doing division before brackets) because they figure the 8/2 is "outside the brackets", but in fact only the 2 is, because the slash separates them as being 2 terms.
It doesn’t have to be confusing. This particular formula is presented in a confusing way. Written differently, the ambiguity is easily resolved.
No, it isn't. You just have to obey all the relevant rules of Maths
PEMDAS
Parenthesis, exponents, multiplication, division, addition, subtraction.
The rule is much older than me and they taught it in school. Nothing ambiguous about it, homie. The phone app is fucked up. Calculator nailed it.
Left to right. If you’re following ALL of the rules of PEMDAS then the answer is 16
Multiplication before division bro.
The division goes last.
Answer is 1.
The answer is 1, but the logic you've used to get there is a little off. Different groups actually follow different logic, but they usually arrive at the same end-point.
The American Mathematical Society goes:
While the American Physical Society does
In both cases, addition and subtraction are equal in priority (this solves the problem brought up by a different comment where following primary school BIDMAS would mean 8-4+2=2). In one case (and this is the way I prefer to do it) they solve the problem by declaring that implicit multiplication is done before division, but explicit multiplication with the × sign follows the same rules you would have learnt in primary school. The other says all multiplication is done before division, including explicit multiplication.
It's never that - that's how people are getting the wrong answer. In high school you get taught about The Distributive Law.
Multiplication and division have the same priority, whichever one comes first LTR is the one that gets resolved first, so it's (8 / 2) * 4
By that logic: 8-2+4=2
Of course, it could be kind of ambiguous, but typical convention gives multiplication/division the same priority, as it does addition/subtraction.
And in general, you need to go left to right when dealing with division and subtraction, if other operations have the same priority.
Right answer but wrong reason. It's because distribution comes first
The comment from subignition explains that the phone's answer, 16, is what you get by strictly following PEMDAS: the rule is that multiplication and division have the same precedence, and you evaluate them from left-to-right.
The calculator uses a different convention where either multiplication has higher priority than division, or where "implicit" multiplication has higher priority (where there is no multiply sign between adjacent expressions).
But explicit multiplication is part of the parenthesis, so still comes before division or exponent
The parentheses step only covers expressions inside parentheses. That's 2 + 2 in this case. The times-2 part is outside the parentheses so it's evaluated in a different step.
except 1 is what you get from strictly following PEMDAS. If you got 16 then you missed one of more rules.
Go back and read your link again. You'll find they're obeying The Distributive Law. i.e. solve all brackets first, from inner-most out.
There's no such thing as implicit multiplication
i know about pemdas and also my brother in christ half the people in the comments are saying the phone app is right lmao
edit: my first answer was 16
All the Maths textbooks agree