Glitch in the matrix

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.
- you could easily make this more straightforward by putting parentheses around 8÷2
  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!").
write ambigous terms like this
It's not ambiguous
all the people in the comments who weren’t educated properly on what conventions are
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
  undefined
  
  1 - x 2
  to make very clear what it's doing.
  
  Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted
  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.
  
  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.
  
  Mathematicians know wolfram is 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.
  like claiming the world isn’t definitely round because some people argue its flat
  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.
  Sometimes people are 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.
  
  they don’t typically write division inline like that
  Yes we do.
  But if you want to argue that Wolfram-Alpha’s equation parser is wrong go ahead
  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.
  
  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.
math is literally the only subject that has rules set in stone
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
- go past past high school and this isn’t remotely true
  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 of math are not set in stone
  The rules around order of operations are!
  if your notation is ambiguous or unclear to your audience try to fix it
  Nothing ambiguous in this expression.
math is literally the only subject that has rules set in stone
Indeed, it does.
This example is specifically made to cause confusion.
No, it isn't. It simply tests who has remembered all the rules of Maths and who hasn't.
Division has the same priority as multiplication
And there's no multiplication here - only brackets and division (and addition within the brackets).
A fraction could be writen up as (x)/(y) not x/y
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.
The fact that some people argue that you do () first and then do what’s outside it means that
...they know all the relevant rules of Maths
look up the facts for yourself
You can find them here
your comment is just as incorrect as everyone who said the answer is 1
and 1 is 100% correct.
well they don’t agree on 0^0
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.
  This example is specifically made to cause confusion.
  No, it isn't. It simply tests who has remembered all the rules of Maths and who hasn't.
  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. If x = 2, it is:
  4/2*2 = 2*2 = 4
  4/(2*2) = 4/4 = 1
  Wolfram solved this with going with the second if it is an X or another variable as it's more intuitive.
  Division has the same priority as multiplication
  And there's no multiplication here - only brackets and division (and addition within the brackets).
  Are you sure ur not a troll? how do you calculate 2(1+1)? It's 4. 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.
  A fraction could be writen up as (x)/(y) not x/y
  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.
  I don't even fully understand you here. If we have a faction; at the top we have 1+2 and at the bottom we have 6-3. inline we could write this as (1+2)/(6-3). The result is 1 as if we simplify it's 3/3.
  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.
  The fact that some people argue that you do () first and then do what’s outside it means that
  ...they know all the relevant rules of Maths
  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? 2x is the same as 2*x and that's the same as 2(x).
  look up the facts for yourself
  You can find them here
  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!
  your comment is just as incorrect as everyone who said the answer is 1
  and 1 is 100% correct.
  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 confusion
  8/2*(2+2), brackets
  8/2*(4), mult & div, left -> right
  4*(4), let go
  4*4, the only
  16, answer
  BUT, and as I stated above IF it'd be like: 8/2x with x=2+2 then, we kinda decided to put implicit brackets there so it's more like 8/(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.
  well they don’t agree on 0^0
  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).
  You said it yourself, if we lim (x->0) y/x then there is an answer. But we aren't in limits. x/0 in 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.
  
  I’m talking about the classic 4/2x. If x = 2, it is:
  4/2x2 = 2x2 = 4
  4/(2x2) = 4/4 = 1
  It's the latter, as per the definition of Terms. There are references to this definition being used going back more than 100 years.
  Wolfram solved this with going with the second if it is an X or another variable as it’s more intuitive
  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).
  how do you calculate 2(1+1)? It’s 4. It’s called implicit multiplication
  No, it's not called implicit multiplication. It's distribution.
  We could write this up as +2*(+1+(+1))
  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.
  I can’t think of a situation where it’s incorrect
  If a=2 and b=3, then...
  1/axb=3/2
  1/ab=(1/6)
  If there is a letter or number or anything next to a bracket, it’s multiplication
  No, it's distribution. Multiplication refers literally to multiplication signs, of which there aren't any in this expression.
  
  2x is the same as 2*x
  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)
  This guy talks in hashtags.
  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. :-)
  Grab a calculator, look at wolfram docs, ask a professor or teacher
  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. :-)
  8/2(2+2), let’s remove the confusion
  8/2*(2+2), brackets
  8/2*(4), mult & div, left -> right
  4*(4), let go
  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.
  it’s more like 8/(2x), but it’s just harder to read, so we don’t
  But that's exactly what we do (but no extra brackets needed around 2x nor 2(2+2) - each is a single term).
  you could argue that it should be handled like the scenario with the x
  Which is what the rules of Maths tells us to do - treat a single term as a single term. :-)
  there are no rules of this
  Yeah, there is. :-)
  you use fractions instead of inline division
  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.
  And by all means, correct me if I’m wrong
  Have done, and appreciate the proper conversation (as opposed to those who call me names for simply pointing out the actual rules of Maths).
  link something that isn’t an unreadable
  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.
  as long as we are civilised
  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, = 64
  but it's just simply incorrect.
  2(4)^2, wow we're at a 2x^2
  2*16 = 32
  The 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+5 we call 2x^2 and 3x and 5 terms. 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.
  
  because brackets are leftmost you do them first
  No, not because leftmost (did I say leftmost? No, I did not), because brackets. Brackets are always first in order of operations.
  2(4)^{2, wow we’re at a 2x}2
  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.
  all sorts of properties. But they are not rules
  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).
  they only apply when we have unknows
  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).
  
  terms get prio because they are terms!?
  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
  
  There are no mention of term prio in the book.
  Which book? I don't know what you're talking about now.
  we have a simplified expression
  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.
  2x^{2+3x+5 we call 2x}2 and 3x and 5 terms. And yes they get priority, not because we named them those, but because they are multiplications
  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.
  using terms, as we just get a single number
  EXACTLY!! When a=2 and b=3, ab=6, a single number. AKA a Term.
  I totally understand why someone would use this, it’s easier
  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.
  I forgot the name/keywords but if you read a calculator’s manual there must be a chapter or something regarding this exact issue.
  The name is Term. You can read about this exact issue in Maths textbooks.
  Especially if you teach physics
  I teach Maths, on which much of Physics is built.
  As for your sources, you still linked a blog post
  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=64
  You 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=32
  or y'know 2*16=32
  Maybe 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= 32
  I 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.
  
  2(4)2=(2x4)2=82=64
  Yes, that's right. Brackets before Exponents, as per the order of operations rules.
  You can’t distribute into a bracket
  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).
  wolfram syntax
  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.
  If you respond with more bullshiting,
  You call quoting Maths textbooks "bullshiting"?

It doesn’t have to be confusing. This particular formula is presented in a confusing way. Written differently, the ambiguity is easily resolved.

This particular formula is presented in a confusing way
No, it isn't. You just have to obey all the relevant rules of Maths

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.

implicit multiplication
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.
  
  Something about the way this thread was written was kind of confusing,
  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
  Is it just that the terminology is wrong? Or am I missing something?
  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.
  a x b, a*b, ab, and a(b) are all acceptable notations to describe the operation “multiply a and b.”
  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.
  
  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
  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)
  
  8/2(1+3) even if they technically are meant to be evaluated the same
  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.
  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
  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).
  if I was factoring something
  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.
  I’m just being a bit lose with the terminology
  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!
  this can rapidly get unreadable once you nest more than a few parens,
  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!
  (8)/(2(1+3)) is obviously different than (8/2)(1+3)
  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.

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:
  Brackets
  Indices
  Multiplication indicated by juxtaposition
  Regular multiplication and division
  Addition and subtraction
  While the American Physical Society does
  Brackets
  Indices
  Multiplication
  Division
  Addition and subtraction
  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.
  
  follows the same rules you would have learnt in primary school
  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
  
  it’s (8 / 2) * 4
  It's 8/2(2+2)=8/(4+4) or 8/2(2+2)=8/2(4)=8/(2x4)
  
  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
- If you’re following ALL of the rules of PEMDAS then the answer is
  ...1. If you got 16 then there's one or more rules that you didn't obey.
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.
  
  The parentheses step only covers expressions inside parentheses
  No, it doesn't
- 16, is what you get by strictly following PEMDAS
  except 1 is what you get from strictly following PEMDAS. If you got 16 then you missed one of more rules.
  the rule is that multiplication and division have the same precedence, and you evaluate them from left-to-right
  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.
  “implicit” multiplication
  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

rules nobody can even agree on!

All the Maths textbooks agree

8

ax + a