1y ago

Glitch in the matrix

574 comments

watches the people with basic math skills fight to the death over the answer
- If you really wanna see a bloodbath, watch this:
  You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
  
  50%, since the coins are independent, right?
  
  This is basically Monty Hall right? The other child is a girl with 2/3 probability, because the first one being a boy eliminates the case where both children are girls, leaving three total cases, in two of which the other child is a girl (BG, GB, BB).
  
  Cheeky bastard.
  It is 50-50, though. The remaining possible states are BG and BB. Both are equally likely. Any further inference is narrative... not statistics.
  The classic example of this is flipping 100 coins. If you get heads 99 times in a row... the last coin is still 50-50. Yes, it is obscenely unlikely to get heads 100 times in a row. But it's already obscenely unlikely to get heads 99 times in a row. And it is obscenely unlikely to alternate perfectly between heads and tails. And it is obscenely unlikely to get a binary pattern spelling out the alphabet. And it is obscenely unlikely to get... literally any pattern.
  Every pattern is equally unlikely, with a fair coin. We see 99 heads in a row versus 1 tails at the end, and think it narrowly averted the least-probable outcome. But only because we lump together all sequences with exactly one tails. That's one hundred different patterns. 1-99 is not the same as 99-1. We just treat them the same because we fixate on uniformity.
  Compare a non-binary choice: a ten-sided die. Thirty 1s in a row is about as unlikely as 100 heads in a row. But 1 1 1... 2 is the same as 1 1 1... 3. Getting the first 29 is pretty damn unlikely. One chance in a hundred million trillion. But the final die can land on any number 1-10. Nine of them upset the pattern our ape brains want. Wanting it doesn't make it any more likely. Or any less likely.
  It would be identically unlikely for a 10-sided die to count from 1 to 10, three times in a row. All the faces appear equally. But swap any two events and suddenly it doesn't count. No pun intended.
  If this couple had eight children, for some god-forsaken reason, and you saw seven boys, the eighth kid being another boy is not less likely for it. The possibility space has already been reduced to two possibilities out of... well nine, I suppose, if order doesn't matter. They could have 0-8 boys. They have at least 7. The only field that says the last kid's not a coin toss is genetics, and they say this guy's chromosome game is strong.
  
  i hate it when ppl do nb erasure for their stupid math text problems. use anything else pls
  
  And don't forget that there's always a slim chance that no matter the gender, the other child is GOAT.
  
  Two more for funsies! I flipped two coins. At least one of them landed on heads. What is the probability that both landed on heads? (Note: this is what my comment originally said before I edited it)
  I have two children. At least one of them is a boy born on a Tuesday. What is the probability that I have two boys?

Different compilers have robbed me of all trust in order-of-operations. If there's any possibility of ambiguity - it's going in parentheses. If something's fucky and I can't tell where, well, better parenthesize my equations, just in case.
- This is best practice since there is no standard order of operations across languages. It's an easy place for bugs to sneak in, and it takes a non-insignificant amount of time to debug.
  
  there is no standard order of operations across languages
  Yes there is. The rules of Maths are universal.
  It’s an easy place for bugs to sneak in
  But that's because of programmers not checking the rules of Maths first.
- This is the way. It's an intentionally ambiguously written problem to cause this issue depending on how and where you learned order of operations to cause a fight.
  
  intentionally ambiguously written
  #MathsIsNeverAmbiguous
  learned order of operations to cause a fight
  The order of operations are the same everywhere. The fights arise from people who don't remember them.
- Different compilers
  Different programmers.
  it’s going in parentheses
  Unfortunately some places don't care where you've put brackets, they'll just go ahead and change it anyway. Welcome to my quest to educate.
  
  That's the same ambiguity, numbnuts. Your added parentheses do nothing. If you wanted to express the value 8 over the value 2*(1+3), you should write 8/(2*(1+3)). That is how you eliminate other valid interpretations.
  As illustration of why there are competing valid interpretations: what human being is going to read "8/2 * (1+3)" as anything but 4*4? Those spaces create semantic separation. But obviously most calculators don't have a spacebar, any more than they have to ability to draw a big horizontal line and place 2(1+3) underneath it. Ambiguous syntax for expressing mathematics is not some foundation-shaking contradiction. It's a consequence of limitations in how we express even the most concrete ideas.
  "The rules of math" you keep spamming about are not mathematical proofs - they're arbitrary decisions made by individuals and organizations. In many cases the opposite choice would be equally sensible. Unlike the innate equivalence of multiplication and division, where dividing by two and multiplying by half are interchangeable. Same with addition and subtraction.
  Do you want to argue that 8 - (2) + (1+3) should be 2?

There's quite a few calculators that get this wrong. In college, I found out that Casio calculators do things the right way, are affordable, and readily available. I stuck with it through the rest of my classes.
- Casio does a wonderful job, and it's a shame they aren't more standard in American schooling. Texas Instruments costs more of the same jobs, and is mandatory for certain systems or tests. You need to pay like $40 for a calculator that hasn't changed much if at all from the 1990's.
  Meanwhile I have a Casio fx-115ES Plus and it does everything that one did, plus some nice quality of life features, for less money.
  
  $40??!! My ti that was required was like over $200!!
  
  TI did the same thing Quark and Adobe did later on – got dominance in their markets, killed off their competition, and then sat back and rested on their laurels thinking they were untouchable
  EDIT: although in part, we should thank TI for one thing – if they hadn’t monopolized the calculator market, Commodore would’ve gone into calculators instead of computers
  
  If you're lucky, you can find these TI calculators in thrift shops or other similar places. I've been lucky since I got both of my last 2 graphing calculators at a yard sale and thrift shop respectively, for maybe around $40-$50 for both.
  
  The TI equivalent to the Casio fx-115ES Plus is the TI-36X Pro, and they both cost $20 at Walmart.
- My Casio calculators get this wrong, even the newer ones. BTW the correct answer is 16, right?
  
  16 is the right answer if you use PEMDAS only: (8 ÷ 2) × (2 + 2)
  1 is the right answer if you use implicit/explicit with PEMDAS: 8 ÷ (2 × (2 + 2))
  both are correct answers (as in if you don’t put in extra parentheses to reduce ambiguity, you should expect expect either answer)
  this is also one of the reasons why postfix and prefix notations have an advantage over infix notation
  postfix (HP, RPN, Forth): 2 2 + 8 2 ÷ × .
  prefix (Lisp): (× (÷ 8 2) (+ 2 2))
  
  Yes
  8 / 2 (2+2)
  8 / 2 (4)
  4 (4)
  16
  
  the correct answer is 16, right?
  No, the correct answer is 1.
  
  Depends on the system you use. Most common system worldwide and in the academic circles (the oldest of the two) has 1 as the answer.
- Sharp as well.
- Ditto for Sharp. It's really only Texas Instruments that is the ongoing exception to the rule.

In some countries we're taught to treat implicit multiplications as a block, as if it was surrounded by parenthesis. Not sure what exactly this convention is called, but afaic this shit was never ambiguous here. It is a convention thing, there is no right or wrong as the convention needs to be given first. It is like arguing the spelling of color vs colour.
- This is exactly right. It's not a law of maths in the way that 1+1=2 is a law. It's a convention of notation.
  The vast majority of the time, mathematicians use implicit multiplication (aka multiplication indicated by juxtaposition) at a higher priority than division. This makes sense when you consider something like 1/2x. It's an extremely common thing to want to write, and it would be a pain in the arse to have to write brackets there every single time. So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
  The same logic is what's used here when people arrive at an answer of 1.
  If you were to survey a bunch of mathematicians—and I mean people doing academic research in maths, not primary school teachers—you would find the vast majority of them would get to 1. However, you would first have to give a way to do that survey such that they don't realise the reason they're being surveyed, because if they realise it's over a question like this they'll probably end up saying "it's deliberately ambiguous in an attempt to start arguments".
  
  The real answer is that anyone who deals with math a lot would never write it this way, but use fractions instead
  
  So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
  Sorry but both my phone calculator and TI-84 calculate 1/2X to be the same thing as X/2. It's simply evaluating the equation left to right since multiplication and division have equal priorities.
  X = 5
  Y = 1/2X => (1/2) * X => X/2
  Y = 2.5
  If you want to see Y = 0.1 you must explicitly add parentheses around the 2X.
  Before this thread I have never heard of implicit operations having higher priority than explicit operations, which honestly sounds like 100% bogus anyway.
  You are saying that an implied operation has higher priority than one which I am defining as part of the equation with an operator? Bogus. I don't buy it. Seriously when was this decided?
  I am no mathematics expert, but I have taken up to calc 2 and differential equations and never heard this "rule" before.
  
  It’s not a law of maths in the way that 1+1=2 is a law
  Yes it is, literally! The Distributive Law, and Terms. Also 1+1=2 isn't a Law, but a definition.
  So 1/2x is universally interpreted as 1/(2x)
  Correct, Terms - ab=(axb).
  people doing academic research in maths, not primary school teachers
  Don't ask either - this is actually taught in Year 7.
  if they realise it’s over a question like this they’ll probably end up saying “it’s deliberately ambiguous in an attempt to start arguments”
  The university people, who've forgotten the rules of Maths, certainly say that, but I doubt Primary School teachers would say that - they teach the first stage of order of operations, without coefficients, then high school teachers teach how to do brackets with coefficients (The Distributive Law).
- BDMAS bracket - divide - multiply - add - subtract
  
  BEDMAS: Bracket - Exponent - Divide - Multiply - Add - Subtract
  PEMDAS: Parenthesis - Exponent - Multiply - Divide - Add - Subtract
  Firstly, don't forget exponents come before multiply/divide. More importantly, neither defines wether implied multiplication is a multiply/divide operation or a bracketed operation.
  
  afair, multiplication was always before division, also as addition was before subtraction
  
  I will never forget this.
- I think when a number or variable is adjacent a bracket or parenthesis then it's distribution to the terms within should always take place before any other multiplication or division outside of it. I think there is a clear right answer and it's 1.
  
  No there is no clear right answer because it is ambiguous. You would never seen it written that way.
  Does it mean A÷[(B)(C)] or A÷B*C
  
  It's 16, addition in bracket comes first
- Not sure what exactly this convention is called
  It's 2 actual rules of Maths - Terms and The Distributive Law.
  never ambiguous
  Correct.
  there is no right or wrong
  Yes there is - obeying the rules is right, disobeying the rules is wrong.
- Not sure what exactly this convention is called
  It's The Distributive Law
  It is a convention thing, there is no right or wrong
  No, it's an actual rule, and 1 is the only correct answer here - if you got 16 then you didn't obey the rule.

Please Excuse My Dear Aunt Sally, she downloaded a shitty ad-infested calculator from the Google Play store.
- Unfortunately, it's the best calculator I could find so far (for my own needs). I paid to remove the ads though, ads bother me way too much to use something infested with them.
  
  If you're willing to pirate (or legally generate) a TI calculator ROM, then Graph 89 is probably what you're looking for. This is what I use as my daily driver calculator with a TI-89 ROM.
  
  wabbitemu!!! Its literally a ti emulator

[...] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
https://people.math.harvard.edu/knill/pedagogy/ambiguity/index.html
As youngsters, math students are drilled in a particular
convention for the "order of operations," which dictates the order thus:
parentheses, exponents, multiplication and division (to be treated
on equal footing, with ties broken by working from left to right), and
addition and subtraction (likewise of equal priority, with ties similarly
broken). Strict adherence to this elementary PEMDAS convention, I argued,
leads to only one answer: 16.
Nonetheless, many readers (including my editor), equally adherent to what
they regarded as the standard order of operations, strenuously insisted
the right answer was 1. What was going on? After reading through the
many comments on the article, I realized most of these respondents were
using a different (and more sophisticated) convention than the elementary
PEMDAS convention I had described in the article.
In this more sophisticated convention, which is often used in
algebra, implicit multiplication is given higher priority than explicit
multiplication or explicit division, in which those operations are written
explicitly with symbols like x / or ÷. Under this more sophisticated
convention, the implicit multiplication in 2(2 + 2) is given higher
priority than the explicit division in 8÷2(2 + 2). In other words,
2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1.
By the same rule, many commenters argued that the expression 8 ÷ 2(4)
was not synonymous with 8÷2x4, because the parentheses demanded immediate
resolution, thus giving 8÷8 = 1 again.
This convention is very reasonable, and I agree that the answer is 1
if we adhere to it. But it is not universally adopted.
- Everyone in this threading referencing PEMDAS and still thinking the answer is 1 are completely ignoring the part of the convention is left to right. Only way to get 1 is to violate left to right on multiplication and division.
  
  The problem is that BIDMAS and its variants are lies-to-children. Real mathematicians don't use BIDMAS. Multiplication by juxtaposition is extremely common, and always takes priority over division.
  Nobody in their right minds would saw 1/2x is the same as (1/2)x. It's 1/(2x).
  That's how you get 1. By following conventions used by mathematicians at any level higher than primary school education.
  
  Only way to get 1 is to violate left to right on multiplication and division
  Actually the only way to get 16 is to ignore one of more rules of Maths - sometimes it's Terms, sometimes it's The Distributive Law, but always something. If you follow all the rules of Maths you get 1.
- That's cool, but still wrong :3
- Exactly, explicit multiplication is part of the parenthesis so it comes first in order of operations
- […] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
  https://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html
  Yeah nah. Actual Maths textbooks and proofs - did you not notice the complete lack of references to textbooks in the blog? It's funny that he mentions Cajori though, given Cajori has a direct reference to Terms #MathsIsNeverAmbiguous
  
  I think I'm gonna trust someone from Harvard over your as-seen-on-TV looking ass account, but thanks for the entertainment you've provided by trying to argue with some of the actual mathematicians in here

I'm with the right answer here. / and * have same precedence and if you wanted to treat 2(2+2) as a single unit, you should have written it like (2*(2+2)).
- It's pretty common even in academic literature to treat implied multiplication as having higher precedence than explicit multiplication/division. Otherwise an expression like 1 / 2n would have to be interpreted as (1 / 2) * n rather than the more natural 1 / (2 * n).
  A lot of this bullshit can be avoided with better notation systems, but calculators tend to be limited in what you can write, so meh. Unless you want to mislead people for the memes, just put parentheses around things.
  
  That's fair. Personally, I just have a grudge against math notation in general. Makes my programmer brain hurt when there's no consistency and a lot of implicit rules.
  Then again, I also like Lisp so I'm not exactly without sin.
  
  The problem is whether or not that rule is taught depends on when and where you learned it. Schools only started teaching that rule relatively recently, and even then, not universally. Which of course makes for ideal engagement bait on your hellsite of choice.
- I’m with the right answer here
  Apparently not.
  if you wanted to treat 2(2+2) as a single unit
  Yes, it is a Term subject to The Distributive Law, written just the way it is.

People keep debating over this stuff. I have a simpler solution. Math is not real.
- The only real answer lmao. People really out here thinking the funny symbols on the paper follow absolute laws. Crazy.
  
  thinking the funny symbols on the paper follow absolute laws
  They do. Maths is universal, just like the laws of Physics (which are often written using Maths BTW).
- My mom's a mathematician, she got annoyed when I said that the order of operations is just arbitrary rules made up by people a couple thousand years ago
  
  It's organized so that more powerful operations get precedence, which seems natural.
  Set aside intentionally confusing expressions. The basic idea of the Order of Operations holds water even without ever formally learning the rules.
  If an addition result comes first and gets exponentiated, the changes from the addition are exaggerated. It makes addition more powerful than it should be. The big stuff should happen first, then the more granular operations. Of course, there are specific cases where we need to reorder, or add clarity, which is why human decisions about groupings are at the top.
  
  My mom’s a mathematician, she got annoyed when I said that the order of operations is just arbitrary rules made up by people a couple thousand years ago
  I'm not surprised. Here's the proof of the order of operations rules. Also, the equals sign wasn't invented until the 16th century, so only 500 years old at most (the earliest references to order of operations are over 400 years ago).
- I'm with you. Has anyone ever actually seen a math? Can you buy a math at the math store? Are there bespoke math craftspeople?
  No.
  I rest my case.
  
  Is math in the room with us right now?
  
  "Math" is a mass noun. You can't have "a math". It's like blood or love. You can have more blood or less blood. There might be units in which blood is measured that you can have a certain number of ("a gallon of blood"), but you can't have, unqualified, a blood or two bloods (well, not in that sense of the word, anyway).

this is why I never use ÷ (or more realistically "/") without explicit brackets denoting order of operations.
- Yeah, if there's any ambiguity, you probably should have written it in a different way.

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.
  
  Huge shout out to the jaded AF high school math teachers that don't give a fuck any more!
  
  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".
  
  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
  
  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.
  
  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).
- 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
- 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
- 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
  
  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).
  
  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

Just write it out as a fraction and avoid all the confusion

The calculator is correct

People in this thread need to watch this: https://youtu.be/lLCDca6dYpA
- Here is an alternative Piped link(s):
  https://piped.video/lLCDca6dYpA
  Piped is a privacy-respecting open-source alternative frontend to YouTube.
  I'm open-source; check me out at GitHub.
- And the much longer video by the same person:
  Problem with PEMDAS: Why Calculators Disagree https://youtu.be/4x-BcYCiKCk
  
  Problem with PEMDAS: Why Calculators Disagree https://youtu.be/4x-BcYCiKCk
  Debunked here - she never once refers to an actual Maths textbook!
- I couldn't listen, voice way too off-putting
  
  Really?! I find her voice incredibly sexy.
- This is the best video out there. A lot of people in north america have no idea.
  
  This is the best video out there
  Debunked here - she never once refers to an actual Maths textbook!
- People in this thread need to watch this: https://youtu.be/lLCDca6dYpA
  Debunked here. She managed to never once refer to an actual Maths textbook! Spoiler alert: everyone who has claimed it's "ambiguous" has done the same thing - no references to any Maths textbooks.
  
  If you think I'm navigating that mess of cross linked posts, well, you're in for a surprise.
  You're really late to this thread.
  She didn't reference any math textbooks because she made the video for commoners, aka not math majors. Her explanations make sense even if they're technically wrong from the perspective of pure mathematics.
  Unfortunately, I don't think many people are going to see your reply, and fewer still will deal with the format you've chosen to present it in; an even smaller subset will likely understand the concepts you're trying to explain.
  Unfortunately, posting this, so long after the thread was active, linking to your own social media as a reference, seems a lot more like attention seeking behavior. The kind of thing I would expect from a bot or phishing attack, especially since you seem to have copy/pasted the reply on several comments. It's like you searched for the YouTube link and just vomitted the same reply on every reference to it. That's bot behavior.
  I'm not saying you're actually a bot, or that anything you've posted is incorrect at all. It just seems suspect.
  
  Here is an alternative Piped link(s):
  https://piped.video/lLCDca6dYpA
  Piped is a privacy-respecting open-source alternative frontend to YouTube.
  I'm open-source; check me out at GitHub.

The problem is that there's no "external" parentheses to really tell us which is right: (8 / 2) * 4 or 8 / (2 * 4)
The amount of comments here shows how much debate this "simple" thing generates
- When there are no parentheses, you process left to right on the same tier of operations. That's how it's always been processed.
- Afaik the order of operations doesn't have distributive property in it. It would instead simply become multiplication and would go left to right and would therefore be 16.
  
  Afaik the order of operations doesn’t have distributive property in it
  The Distributive Law applies to all bracketed terms that have a coefficient. It's literally the first step in solving brackets.
- If you agree that parenthesis go first then the equation becomes 8/2x4. Then it's simply left to right because multiplication does not take precedence over division. What's the nuanced talk? That M comes before D in PEMDAS?
  
  Finally someone in here who knows math. Thank you.
  
  My observation was mainly based on this other comment
  https://programming.dev/comment/5414285
  In this more sophisticated convention, which is often used in algebra, implicit multiplication is given higher priority than explicit multiplication or explicit division, in which those operations are written explicitly with symbols like x / or ÷. Under this more sophisticated convention, the implicit multiplication in 2(2 + 2) is given higher priority than the explicit division in 8÷2(2 + 2). In other words, 2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1. By the same rule, many commenters argued that the expression 8 ÷ 2(4) was not synonymous with 8÷2x4, because the parentheses demanded immediate resolution, thus giving 8÷8 = 1 again.
  
  If you agree that parenthesis go first then the equation becomes 8/2x4
  No, it becomes 8/(2x4). You can't remove brackets unless there's only 1 term left inside. Removing them prematurely flips the 4 from being in the denominator to being in the numerator, hence the wrong answer.
- The problem is that there’s no “external” parentheses to really tell us which is right: (8 / 2) * 4 or 8 / (2 * 4)
  The Distributive Law tells us it's the latter.

For anyone like me who has math as their worst subject: PEMDAS.
PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.
So we gotta do it in the proper order. And remember, if the number is written like 2(3) then its multiplication, as if it was written 2 x 3 or 2 * 3.
So we read 8/2(2+2) and need to do the following;
Read the Parentheses of (2 + 2) and follow the order of operations within them, which gets us 4.
Then we do 2(4) which is the same as 2 x 4 which is 8
8 / 8 is 1.
The answer is 1. The old calculator is correct, the phone app which has ads backed into it for a thing that all computers were invented to do is inaccurate.
- Well that's just wrong... Multiplication and division have equal priorities so they are done from left to right. So: 8 / 2 * (2 + 2)=8 / 2 * 4=4 * 4=16
  
  This but unironically
  
  Implicit multiplication takes priority over explicit multiplication or division. 2(2+2) is not the same thing as 2*(2+2).
  
  Not quite, pemdas can go either from the left or right (as long as you are consistent) and division is the same priority as multiplication because dividing by something is equal to multiplying by the inverse of that thing... same as subtraction being just addition but you flip the sign.
  8×1/2=8/2 1-1=1+(-1)
  The result is 16 if you rewrite the problem with this in mind: 8÷2(2+2)=8×(1/2)×(2+2)
  
  8 / 2 * (2 + 2)
  That's not the same as 8 / 2 (2 + 2). In the original question, 2(2+2) is a single term in the denominator, when you added the multiply you separated it and thus flipped the (2+2) to be in the numerator, hence the wrong answer.
- Uh.. no the 1 is wrong? Division and multiplication have the same precedence, so the correct order is to evaluate from left to right, resulting in 16.
  
  The real correct order is to use brackets to remove ambiguity.
  
  There's no multiplication in this question - multiplication refers literally to multiplication signs - only division and brackets, and addition within the brackets. So you have to use The Distributive Law to solve the brackets, then do the division, giving you 1.
- not to be That Guy, but the phone is actually correct... multiplication and division have the same precedence, so 8 / 2 * 4 should give the same result as 8 * 4 / 2, ie 16
  
  but the phone is actually correct
  No, it's actually wrong.
  8 / 2 * 4
  It's 8/(2x4). You can't remove brackets unless there is only 1 term left inside.
- The problem with this is that the division symbol is not an accurate representation of the intended meaning. Division is usually written in fractions which has an implied set of parenthesis, and is the same priority as multiplication. This is because dividing by a number is the same as multiplying by the inverse, same as subtracting is adding the negative of a number.
  8/2(2+2) could be rewritten as 8×1/2×(2+2) or (8×(2+2))/2 which both resolve into 16.
  
  You left out the way it can be rewritten which most mathematicians would actually use, which is 8/(2(2+2)), which resolves to 1.
  
  Division is usually written in fractions
  Division and fractions aren't the same thing.
  fractions which has an implied set of parenthesis
  Fractions are explicitly Terms. Terms are separated by operators (such as division) and joined by grouping symbols (such as a fraction bar), so 1÷2 is 2 terms, but ½ is 1 term.
  8/2(2+2) could be rewritten as 8×1/2×(2+2)
  No, it can't. 2(2+2) is 1 term, in the denominator. When you added the multiply you broke it into 2 terms, and sent the (2+2) into the numerator, thus leading to a different answer. 8/2(2+2)=1.
- P E M D A S
  vs
  P E M/D A/S
  The latter is correct, Multiplication/Division, and Addition/Subtraction each evaluate left to right (when not made unambiguous by Parentheses). I.e., 6÷2×3 = 9, not 1. That said, writing the expression in a way that leaves ambiguity is bad practice. Always use parentheses to group operations when ambiguity might arise.
- You are correct. This is the right sequence of operations done here.
- PEMDAS evaluated from left to right. If you followed that you’d get 16. 1 is ignoring left to right.
  
  1 isn't ignoring anything. 16 can be arrived at by ignoring any one of multiple order of operations rules.
- The problem is that the way PEMDAS is usually taught multiplication and division are supposed to have equal precedence. The acronym makes it look like multiplication comes before division, but you're supposed to read MD and as one step. (The same goes for addition and subtraction so AS is also supposed to be one step.) It this example the division is left of the multiplication so because they have equal precedence (according to PEMDAS) the division applies first.
  IMO it's bad acronym design. It would be easier if multiplication did come before division because that is how everyone intuitively reads the acronym.
  Maybe it should be PE(M/D)(A/S). But that version is tricky to pronounce. Or maybe there shouldn't be an acronym at all.
  
  but you’re supposed to read MD and as one step
  You can do them in any order at all - M then D, D then M (hence the acronym BEDMAS), or all in one - what does matter is not treating Distribution as though it's Multiplication (which refers literally to multiplication signs), when in actual fact it's the first step in solving Brackets.
- Ignore the idiots telling you you're wrong. Everyone with a degree in math, science or engineering makes a distinction between implicit and explicit multiplication and gives implicit multiplication priority.
- You're a lifesaver, thank you so much. I actually didn't know about PEMDAS, I was never taught it before...
- Turns out I’m wrong, but I haven’t been told how or why. I’m willing to learn if people actually tell me
  Well, I don't know what you said originally, so I don't know what it is you were told was wrong - 1 or 16? 😂 The correct answer is 1.
  Anyhow, I have an order of operations thread which covers literally everything there is to know about it (including covering all the common mistakes and false claims made by some). It includes textbook references, historical Maths documents, worked examples, proofs, memes, the works! I'm a high school Maths teacher/tutor - I've taught this topic many times.

The correct answer is 16. Multiplication and Division happen at the same level of priority, and are evaluated left-to-right.
- No it's ambiguous, you claiming there is one right answer is actually wrong.
  
  It is not ambiguous at all, there absolutely is one right answer, and it is 16.
  
  But there actually is only 1 right answer, and unfortunately for the person you're replying to it's 1.
- PEMDAS be damned?
  
  PEMDAS should be read as Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. There are four levels of priority, not six.

Left is correct; implicit multiplication takes precedence over explicit multiplication or division.
- What the fuck is the difference in implicit vs explicit? It’s the same operation, why the fuck does it matter if there is a symbol?
  Multiplication comes first, then division.
  
  Division is a form of Multiplication, just as subtraction is a form of addition. You multiply and divide in the same step, left to right
  
  No, multiplication and division are resolved from left to right in the same step. But implicit multiplication (xy, as opposed to x*y) happens first.
  
  the difference in implicit vs explicit? It’s the same operation
  "implicit multiplication" isn't even a real thing in Maths, and isn't even multiplication to begin with - people use that umbrella term to either mean The Distributive Law - which is the first step is solving Brackets - or Terms, which are products, which is the result of a multiplication.
  e.g. if a=2 and b=3, then...
  axb=2x3 - 2 terms
  ab=6 - 1 term
  Multiplication comes first, then division
  They can be done in either order, or even together, as long as you go left to right.
- What's the difference between implicit multiplication and explicit multiplication?
  
  Implicit multiplication is xy or x(y), explicit multiplication is x*y.
  Basically just whether or not there's an actual multiplication symbol.

Throw that ecal in the trash

This is exactly why we have Reversed Polish Notation. When will people learn?
- A fifteen year old version of myself somewhere inside just screamed in iptscrae induced frustration.
- RPN Gang unite!

This is why I loved my Casio 2D. It could use actual fractions to avoid these kind of issues

there's a setting in Qalculate! that asks if you want implicit multiplication to apply to the denominator or the numerator

I don’t understand why people say Maths. Math encompasses every single type of Math. Maths is just wrong.
- Because it's British English
  
  That may be, but Math still encompasses all Math so there’s no need to pluralize it.
  
  And British English is wrong. Those motherfuckers stick "u" into way too many words.
- What is math short for?
  
  Mathollomew
  
  Maths stands for
  Mathematical
  Anti
  Telharsic
  Harfatum
  Septomin
  
  Mathmallow
- It's a shortened form of mathematics, build a bridge and get over it
  
  Ok, but the British also shortened television and made it tele. That makes sense because they took part of the word to do it. If you were going to shorten the word mathematics, why wouldn’t it be math, especially when that would follow what you did with television. Why shorten the word and then add the s from the end for no reason?
- I don’t understand why people say Maths
  Because it's plural.
  every single type of Math
  In other words, every branch of Mathematics.

Ah damn it. It took me ages to find a calculator app that fits my needs..... And now I find out it works like the one on the right.
- ... the one on the right is correct.... that's a jank ass calculator on the left that doesn't know how to do order of operations 8/2×(2+2) 8/2x4 4x4 16
  
  There isn't a multiplication symbol though. By your logic something like 8÷2x would mean (8÷2)*x because order of operations
  Or if you read 8÷2√x as (8÷2)*√x
  Just notate 8÷2(2+2) as 8÷2x; x=(2+2) and you get it, you can substitute any complete expression with a variable in an equation and the logic stays the same.
  
  the one on the right is correct
  No, it isn't.
  8/2×(2+2)
  ...isn't the same thing as 8/2(2+2). You separated the term in the denominator, leading the (2+2) to get flipped into the numerator, hence wrong answer.
- Hiper Calc is the calculator app that I use. It's very good. When I ran this equation, it actually notified me how the operands should be grouped (weak or strong) and provided two answers. Honestly the whole issue can be avoided if you use more parentheses
  
  the whole issue can be avoided
  ...by following all the order of operations rules
- HP Prime, king of all. Or if you prefer open source NumWorks.

I don't think you encounter this one very often, but the technically correct -2^2 = -4 has a higher chance of ruining your day.
- You mean x^2 =4 where x=±2
  
  No, you'd expect that -2^{2 would equal 4, but calculators solve it as -(2)}2 not (-2)^2. But the case you mentioned is also pretty common.
- Oh goddammit! Why doesn't PEMDAS prepare us for unary negation??

8÷2(2+2)=2(2+2)÷2(2+2)
alternatively if 8÷2(2+2)=16 that means 2(2+2)=8÷16 in other words 8=0,5 which it isnt
- your first line is correct, but while it looks like 1 (and it might be under different conventions), evaluating according to standard rules (left to right if not disambiguated by pemdas) yields
  2(2+2)/2(2+2) = 2(4)/2(4) = 24/24 = 8/24 = 44 = 16
  Using implicit multiplication in quotients is weird and really shouldn't happen, this would usually be written as 8/(2(2+2)) or 8/2(2+2) and both are much clearer
  Your second argument only works if you treat 2(2+2) as a single "thing", which it looks like, but isn't, in this case
  
  not much to refute in the argument of whether its 16 or 1 as its all a matter of convention in the end and ultimately the root of the argument is poor formatting of the expression, im used to implicit multiplication taking precedent and that 2(2+2)===2*(2+2) and that for my first argument having the same expression on 2 sides of a division sign automatically equals 1, but how come you find implicit multiplication in quotients weird? seeing as it happens literally all the time in equations, unless thats a difference in school systems or similar im unaware of
  for fun also rewrote the expression into powers of 2 and indeed depending on how you go about implicit multiplication i end up with either 2⁰ or 2⁴, so for the sake of sanity i figure its best to just say x₁=1; x₂=16

https://math.berkeley.edu/%7Egbergman/misc/numbers/ord_ops.html
- so far as I know, [BIDMAS] is a creation of some educator, who has taken conventions in real use, and extended them to cover cases where there is no accepted convention. So it misleads students; and moreover, if students are taught PEMDAS by rote without the proviso mentioned above, they will not even get the standard interpretation of a−b+c.
  Absolutely rekt.
- https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
  The rules of Maths that he manages to completely ignore.

I treat • and × differently, • I treat like the left side and × I treat like the right side calculation.
- wtf
- I treat • and × differently, • I treat like the left side and × I treat like the right side calculation
  They literally mean the same thing - just one is used in some countries and the other is used in other countries.
  
  Yes they both are multiply, but...
  Calculate 8 ÷ 2a where a = 4. Then,
  Calculate 8 ÷ 2 × a where a = 4.
  See how in the first form a is implied to be part of the fraction where in the second it isn't?
  A dot • could be between 2 and a and it would still follow the first example. In vector multiplication, dot and cross products produce different results.

So umm what's the correct one?
- Ultimately it's ambiguous and bad. But most mathematicians (at a level higher than primary school education) use multiplication by juxtaposition—aka implicit multiplication—at a higher priority than division. BIDMAS, as you might have been taught in primary school, is an oversimplification that doesn't even account for the possibility of juxtaposition, because you didn't learn about that until secondary school.
  Most mathematicians would use logic that lets you arrive at the answer of "1", while also saying it should have been written better. Brackets are cheap. Use them!
- Please Excuse My Dear Aunt Sally
- If you believe multiplication goes before division then 1. 8 / (2 * 4)
  If you believe multiplication and division are of equal importance 16. 8 / 2 * 4
  SQL, Google, and I believe C++ and HLSL would say the latter: (16)
  
  If you believe multiplication goes before division then 1. 8 / (2 * 4)
  That's not "multiplication" though - that's The Distributive Law. Multiplication refers literally to multiplication signs.
- what’s the correct one?
  Yes, one is correct :-)