this post was submitted on 03 Dec 2023
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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 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.
There is, especially if you're dividing by a fraction! Division and fractions aren't the same thing.
Not if it actually is a division and not a fraction. There's no problem with having multiple divisions in a single expression.
Semantically, yes they are. Syntactically they're different.
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 1hold? No, the results are equal, both are1. How aboutdiv 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.Yes, they would - it's the standard way to write a factorised term.
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
Yes, I'm a Maths teacher.
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.
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.
I can say that this is a common thing in engineering. Pretty much everyone I know would treat 1/2x as 1/(2x).
Which does make it a pain when punched into calculators to remember the way we write it is not necessarily the right way to enter it. So when put into matlab or calculators or what have you the number of brackets can become ridiculous.
I'm an engineer. Writing by hand I would always use a fraction. If I had to write this in an email or something (quickly and informally) either the context would have to be there for someone to know which one I meant or I would use brackets. I certainly wouldn't just wrote 1/2x and expect you to know which one I meant with no additional context or brackets
Yes it is, literally! The Distributive Law, and Terms. Also 1+1=2 isn't a Law, but a definition.
Correct, Terms - ab=(axb).
Don't ask either - this is actually taught in Year 7.
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
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.
~~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
I literally just explained this. The Parenthesis takes priority over multiplication and division outright.
It's 16, addition in bracket comes first
It's 2 actual rules of Maths - Terms and The Distributive Law.
Correct.
Yes there is - obeying the rules is right, disobeying the rules is wrong.
It's The Distributive Law
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.