Glitch in the matrix

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 are you suggesting that Richard Feynman didn't "deal with maths a lot", then? Because there definitely exist examples where he worked within the limitations of the medium he was writing in (namely: writing in places where using bar fractions was not an option) and used juxtaposition for multiplication bound more tightly than division.
  
  Here's another example, from an advanced mathematics textbook:
  
  Both show the use of juxtaposition taking precedence over division.
  I should note that these screenshots are both taken from this video where you can see them with greater context and discussion on the subject.
  
  Mind you, Feynmann clearly states this is a fraction, and denotes it with "/" likely to make sure you treat it as a fraction.
  
  Yep with pen and paper you always write fractions as actual fractions to not confuse yourself, never a division in sight, while with papers you have a page limit to observe. Length of the bars disambiguates precedence which is important because division is not associative; a/(b/c) /= (a/b)/c. "calculate from left to right" type of rules are awkward because they prevent you from arranging stuff freely for readability. None of what he writes there has more than one division in it, chances are that if you see two divisions anywhere in his work he's using fractional notation.
  Multiplication by juxtaposition not binding tightest is something I have only ever heard from Americans citing strange abbreviations as if they were mathematical laws. We were never taught any such procedural stuff in school: If you understand the underlying laws you know what you can do with an expression and what not, it's the difference between teaching calculation and teaching algebra.
  
  never a division in sight
  There is, especially if you're dividing by a fraction! Division and fractions aren't the same thing.
  if you see two divisions anywhere in his work he’s using fractional notation
  Not if it actually is a division and not a fraction. There's no problem with having multiple divisions in a single expression.
  
  Division and fractions aren’t the same thing.
  Semantically, yes they are. Syntactically they're different.
  
  Semantically, yes they are
  No, they're not. Terms are separated by operators (division) and joined by grouping operators (fraction bar).
  
  That's syntax.
  ...let me take this seriously for a second.
  The claim "Division and fractions are semantically distinct" implies that they are provably distinct functions, we can use the usual set-theoretic definition of those. Distinctness of functions implies the presence of pair n, m, elements of an appropriate set, say, the natural numbers without zero for convenience, such that (excuse my Haskell) div n m /= fract n m, where /= is the appropriate inequality of the result set (the rational numbers, in this example, which happens to be decidable which is also convenient).
  Can you give me such a pair of numbers? We can start to enumerate the problem. Does div 1 1 /= fract 1 1 hold? No, the results are equal, both are 1. How about div 1 2 /= fract 1 2? Neither, the results are both the same rational number. I leave exploring the rest of the possibilities as an exercise and apologise for the smugness.
  
  let me take this seriously for a second
  You need to take it seriously for longer than that.
  implies that they are provably distinct functions
  No, I'm explicitly stating they are.
  we can use the usual set-theoretic definition
  This is literally Year 7 Maths - I don't know why some people want to resort to set theory.
  Can you give me such a pair of numbers?
  But that's the problem with your example - you only tried it with 2 numbers. Now throw in another division, like in that other Year 7 topic, dividing by fractions.
  1÷1÷2=½ (must be done left to right)
  1÷½=2
  In other words 1÷½=1÷(1÷2) but not 1÷1÷2. i.e. ½=(1÷2) not 1÷2. Terms are separated by operators (division in this case) and joined by grouping symbols (brackets, fraction bar), and you can't remove brackets unless there is only 1 term left inside, so if you have (1÷2), you can't remove the brackets yet if there's still some of the expression it's in left to be solved (or if it's the last set of brackets left to be solved, then you could change it to ½, because ½=(1÷2)).
  Therefore, as I said, division and fractions aren't the same thing.
  apologise for the smugness
  Apology accepted.
  
  denotes it with “/” likely to make sure you treat it as a fraction
  It's not the slash which makes it a fraction - in fact that is interpreted as division - but the fact that there is no space between the 2 and the square root - that makes it a single term (therefore we are dividing by the whole term). Terms are separated by operators (2 and the square root NOT separated by anything) and joined by grouping symbols (brackets, fraction bars).
  
  used juxtaposition for multiplication bound more tightly than division
  It's called Terms - Terms are separated by operators and joined by grouping symbols. i.e. ab=(axb).
- The real answer is that anyone who deals with math a lot would never write it this way
  Yes, they would - it's the standard way to write a factorised term.
  but use fractions instead
  Fractions and division aren't the same thing.
  
  Fractions and division aren't the same thing.
  Are you for real? A fraction is a shorthand for division with stronger (and therefore less ambiguous) order of operations
  
  Are you for real?
  Yes, I'm a Maths teacher.
  A fraction is a shorthand for division with stronger (and therefore less ambiguous) order of operations
  I added emphasis to where you nearly had it.
  ½ is a single term. 1÷2 is 2 terms. Terms are separated by operators (division in this case) and joined by grouping symbols (fraction bars, brackets).
  1÷½=2
  1÷1÷2=½ (must be done left to right)
  Thus 1÷2 and ½ aren't the same thing (they are equal in simple cases, but not the same thing), but ½ and (1÷2) are the same thing.
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.
- Sorry but both my phone calculator and TI-84 calculate 1/2X
  ...and they're both wrong, because they are disobeying the order of operations rules. Almost all e-calculators are wrong, whereas almost all physical calculators do it correctly (the notable exception being Texas Instruments).
  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?
  The rules of Terms and The Distributive Law, somewhere between 100-400 years ago, as per Maths textbooks of any age. Operators separate terms.
  I am no mathematics expert... never heard this “rule” before.
  I'm a High School Maths teacher/tutor, and have taught it many times.
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.
- neither defines wether implied multiplication is a multiply/divide operation or a bracketed operation
  The Distributive Law says it's a bracketed operation. To be precise "expand and simplify". i.e. a(b+c)=(ab+ac).
- Exponents should be the first thing right? Or are we talking the brackets in exponents..
  
  Exponents are second, parentheses/brackets are always first. What order you do your exponents in is another ambiguity though.
  
  What order you do your exponents in is another ambiguity though
  No it isn't - top down.
  
  2³4 is ambiguous. 2⁽³4) is standard practice, but some calculators aren't that smart and will do (2³⁾4.
  It's ambiguous because it works both ways, not because we don't have a standard. Confusion is possible.
  
  The only confusion I can see is if you intended for the 4 to be an exponent of the 3 and didn't know how to do that inline, or if you did actually intend for the 4 to be a separate numeral in the same term? And I'm confused because you haven't used inline notation in a place that doesn't support exponents of exponents without using inline notation (or a screenshot of it).
  As written, which inline would be written as (2^{3)4, then it's 32. If you intended for the 4 to be an exponent, which would be written inline as 2\}34, then it's 2^81 (which is equal to whatever that is equal to - my calculator batteries are nearly dead).
  we don’t have a standard
  We do have a standard, and I told you what it was. The only confusion here is whether you didn't know how to write that inline or not.
  
  It's ambiguous because it works both ways, not because we don't have a standard.
  Try reading the whole sentence. There is a standard, I'm not claiming there isn't. Confusion exists because operating against the standard doesn't immediately break everything like ignoring brackets would.
  Just to make sure we're on the same page (because different clients render text differently, more ambiguous standards...), what does this text say?
  2³4
  It should say 2^3^4; "Two to the power of three to the power of four". The proper answer is 2⁸¹, but many math interpreters (including Excel, MATLAB, and many students) will instead compute 8⁴, which is quite different.
  We have a standard because it's ambiguous. If there was only one way to do it, we'd just do that, no standard needed. You'd need to go pretty deep into kettle math or group theory to find atypical addition for example.
  
  There is a standard, I’m not claiming there isn’t
  Ah ok. Sorry, got caught out by a double negative in your sentence.
  Confusion exists because operating against the standard doesn’t immediately break everything like ignoring brackets would
  Ah but that's exactly the original issue in this thread - the e-calc is ignoring the rules pertaining to brackets. i.e. The Distributive Law.
  
  Ah ok. Well that was my only confusion was what you had actually intended to write, not how to interpret it (depending on what you had intended). Yes should be interpreted 2^81.
  including Excel
  Yeah, but Excel won't let you put in a factorised term either. It's just severely broken because the people who wrote it didn't bother checking the rules of Maths first. Programmers not knowing the rules of Maths doesn't mean Maths is ambiguous (it certainly creates a lot of confusion though!).
  We have a standard because it’s ambiguous. If there was only one way to do it, we’d just do that,
  Disagree. There is one way to do it - follow the rules of Maths. That's why they exist. The order of operations rules are at least 400 years old, and make it not ambiguous. If people aren't obeying the rules then they're just wrong - that doesn't make it ambiguous. It's like saying if e-calcs started saying 1+1=3 then that must mean 1+1 is ambiguous. It might create confusion, but it doesn't mean the Maths is ambiguous.
  
  Brackets are ALWAYS first.
afair, multiplication was always before division, also as addition was before subtraction
- It's BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
  But multiplication and division are whichever comes first right to left in the expression, and likewise with subtraction.
  Although implicit multiplication is often treated as binding tighter than explicit. 1/2x is usually interpreted as 1/(2x), not (1/2)x.
  
  a fair point, but aren't division and subtraction are non-communicative, hence both operands need to be evaluated first?
  
  It’s commutative, not communicative, btw
  
  whoops, my bad
  
  1 - 3 + 1 is interpreted as (1 - 3) + 1 = -1
  Yes, they're non commutative, and you need to evaluate anything in parens first, but that's basically a red herring here.
  
  ok, i guess you're right
  
  It's BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
  But, since your rule has the D&M as well as the A&S in brackets does that mean your rule means you have to do D&M as well as the A&S in the formula before you do the exponents that are not in brackets?
  But seriously. Only grade school arithmetic textbooks have formulas written in this ambiguous manner. Real mathematicians write their formulas clearly so that there isn't any ambiguity.
  
  That's not really true.
  You'll regularly see textbooks where 3x/2y is written to mean 3x/(2y) rather than (3x/2)*y because they don't want to format
  undefined
  
  3x ---- 2y
  properly because that's a terrible waste of space in many contexts.
  
  You'll regularly see textbooks
  That's what I said.
  
  You generally don't see algebra in grade school textbooks, though.
  
  12 is a grade. I took algebra in the 7th grade.
  
  Grade school is a US synonym for primary or elementary school; it doesn't seem to be used as a term in England or Australia. Apparently, they're often K-6 or K-8; my elementary school was K-4; some places have a middle school or junior high between grade school and high school.
  
  I don't know why you're getting lost on the pedantry of defining "grade school", when I was clearly discussing the fact that you only see this kind of sloppy formula construction in arithmetic textbooks where students are learning the basics of how to perform the calculations. Once you get into applied mathematics and specialized fields that use actual mathematics, like engineering, chemistry and physics, you stop seeing this style of formula construction because the ambiguity of the terms leads directly to errors of interpretation.
- ~~Multiplication VS division doesn't matter just like order of addition and subtraction doesn't matter.. You can do either and get same results.~~
  Edit : the order matters as proven below, hence is important
  
  If you do only multiplication first, then 2×3÷3×2 = 6÷6 = 1.
  If you do mixed division and multiplication left to right, then 2×3÷3×2 = 6÷3×2 = 2×2 = 4.
  Edit: changed whitespace for clarity
  
  4 would be correct since you go left to right.
  
  2nd one is correct, divisions first.
- I was taught that division is just inverse multiplication, and to be treated as such when it came to the order of operations (i.e. they are treated as the same type of operation). Ditto with addition and subtraction.
I will never forget this.
- BDSM Brackets ... ok
- Glad to be of help, I remember it being taughy back in the 4th grade and it stuck well.

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 means
  undefined
  
  A ÷ B(C) which is equivalent to A ÷ (B*C)
  I literally just explained this. The Parenthesis takes priority over multiplication and division outright.
  undefined
  
  Maybe B*C = B(C) But A ÷ B(C) =! A ÷ B * C
  
  No. It's ambiguous. In a math book or written by anyone that actually uses math, you don't have a "%"
  You group stuff below the line, and you use parens and brackets to group things like (a + b) and (x)(y) so that it is not ambiguous.
  2/xy would be almost always interpreted differently than 2/x(x+y) which is ambiguous and could mean (2/x)(x+y) or 2/[(x)(x+y)]
  
  You continue to say it's ambiguous, but the most commonly used convention on earth very clearly prioritizes parenthesis. It is not ambiguous.
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.