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.
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.
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.
I’m not asking you to explain how division isn’t associative, I’m asking you to find an n, m such that ⁿ⁄ₘ is not equal to n ÷ m.
To compare two functions you have to compare them one to one, you have to compare ⁄ to ÷, not two invocation of one to three invocation of the others or whatever.
EDIT: OMG you’re on programming.dev. A programmer should understand the difference between syntax and semantics. This vs. this. Mathematicians tend to call syntax “notation” but it’s the same difference.
Another approach: If frac and div are different functions, then multiplication would have two different inverses. How could that be?
I’m not asking you to explain how division isn’t associative
I was explaining why we have the rule of Terms (which you’ve not managed to find a problem with).
I’m asking you to find an n, m such that ⁿ⁄ₘ is not equal to n ÷ m
I already pointed out that’s irrelevant - it doesn’t involve a division followed by a factorised term. You’re asking me to defend something which I never even said, nor is relevant to the problem. i.e. a strawman designed to deflect.
Stop being confidently incorrect
You haven’t shown that anything I’ve said is incorrect. If you wanna get back on topic, then come back to showing how 1÷½=1÷(1÷2) but not 1÷1÷2 is, according to you, incorrect (given you claimed division and fractions are the same thing)
EDIT: OMG you’re on programming.dev.
Yeah, welcome to why I’m trying to get programmers to learn the rules of Maths
You haven’t shown that anything I’ve said is incorrect. If you wanna get back on topic, then come back to showing how 1÷½=1÷(1÷2) but not 1÷1÷2 is, according to you, incorrect (given you claimed division and fractions are the same thing)
Your statement there is correct. It is also a statement about syntax, not semantics. Divisions and fractions are distinct in syntax, but they still both are the same functions, they both are the inverse of multiplication.
Yeah, welcome to why I’m trying to get programmers to learn the rules of Maths
PEMDAS is not a rule of maths. It’s a bunch of bad American maths pedagogy. Is, in your opinion, “show your work” a rule of maths? Or is it pedagogy?
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.You need to take it seriously for longer than that.
No, I’m explicitly stating they are.
This is literally Year 7 Maths - I don’t know why some people want to resort to set theory.
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.
Apology accepted.
I’m not asking you to explain how division isn’t associative, I’m asking you to find an n, m such that ⁿ⁄ₘ is not equal to n ÷ m.
To compare two functions you have to compare them one to one, you have to compare ⁄ to ÷, not two invocation of one to three invocation of the others or whatever.
Also I’ll leave you with this. Stop being confidently incorrect, it’s a bad habit.
EDIT: OMG you’re on programming.dev. A programmer should understand the difference between syntax and semantics. This vs. this. Mathematicians tend to call syntax “notation” but it’s the same difference.
Another approach: If
fracanddivare different functions, then multiplication would have two different inverses. How could that be?I was explaining why we have the rule of Terms (which you’ve not managed to find a problem with).
I already pointed out that’s irrelevant - it doesn’t involve a division followed by a factorised term. You’re asking me to defend something which I never even said, nor is relevant to the problem. i.e. a strawman designed to deflect.
You haven’t shown that anything I’ve said is incorrect. If you wanna get back on topic, then come back to showing how 1÷½=1÷(1÷2) but not 1÷1÷2 is, according to you, incorrect (given you claimed division and fractions are the same thing)
Yeah, welcome to why I’m trying to get programmers to learn the rules of Maths
I said:
You replied:
Thus, you made a claim about semantics. One which I then went on and challenged you to prove, which you tried to do with a statement about syntax.
I did not say “opposite”. I said “inverse”. That term has a rather precise meaning.
Your statement there is correct. It is also a statement about syntax, not semantics. Divisions and fractions are distinct in syntax, but they still both are the same functions, they both are the inverse of multiplication.
PEMDAS is not a rule of maths. It’s a bunch of bad American maths pedagogy. Is, in your opinion, “show your work” a rule of maths? Or is it pedagogy?
The opposite of div is to multiply. The opposite of frac is to invert the fraction.