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.

[–] [email protected] 1 points 9 months ago (1 children)

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.

[–] [email protected] 3 points 9 months ago (1 children)

Division and fractions aren’t the same thing.

Semantically, yes they are. Syntactically they're different.

[–] [email protected] 1 points 9 months ago (1 children)

Semantically, yes they are

No, they're not. Terms are separated by operators (division) and joined by grouping operators (fraction bar).

[–] [email protected] 2 points 9 months ago (1 children)

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.

[–] [email protected] 1 points 9 months ago (1 children)

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.

[–] [email protected] 2 points 9 months ago* (last edited 9 months ago) (2 children)

1÷½=1÷(1÷2) but not 1÷1÷2.

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 frac and div are different functions, then multiplication would have two different inverses. How could that be?

[–] [email protected] 1 points 9 months ago

If frac and div are different functions, then multiplication would have two different inverses. How could that be?

The opposite of div is to multiply. The opposite of frac is to invert the fraction.

[–] [email protected] 1 points 9 months ago* (last edited 9 months ago) (1 children)

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

[–] [email protected] 2 points 9 months ago (1 children)

You’re asking me to defend something which I never even said

I said:

Semantically, yes they are

You replied:

No, they’re not.

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.

The opposite of div is to multiply. The opposite of frac is to invert the fraction.

I did not say "opposite". I said "inverse". That term has a rather precise meaning.

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?

[–] [email protected] 1 points 9 months ago (1 children)

Thus, you made a claim about semantics

And told you what it was.

challenged you to prove

Which I did with a concrete example, which you have since ignored.

I did not say “opposite”. I said “inverse”

The inverse of div is to multiply. The inverse of frac is to invert the fraction - happy now?

Divisions and fractions are distinct in syntax, but they still both are the same functions

No, they're not. Division is a binary operator, a fraction is a single term.

they both are the inverse of multiplication

Multiplication is also a binary operator, and division is the opposite of it (in the same way that plus and minus are unary operators which are the opposite of each other). A fraction isn't an operator at all - it's a single term. There is no "opposite" to a single term (except maybe another single term which is the opposite of it. e.g. the inverse fraction).

PEMDAS is not a rule of maths

No, it's a mnemonic to remind people of the actual rules.

[–] [email protected] 1 points 9 months ago* (last edited 9 months ago) (1 children)

Which I did with a concrete example, which you have since ignored.

You have given nothing of that sort. You provided a statement about a completely orthogonal topic instead. "Prove that the sky is blue" -- "Here, grass is green" -- "That doesn't answer the question" -- "Nu-uh it does!". That's you. That "Nu-uh".

The inverse of frac is to invert the fraction - happy now?

"Inverting a fraction" is not a functional inverse. You're getting led astray by terminology, those two uses of the word "invert" have nothing to do with each other, it's a case of English having bad terminology (in German we use different terms so the confusion doesn't even begin to apply).

Go read that wikipedia article I linked. Can you even read it. Do you have the necessary mathematical literacy.

No, they’re not. Division is a binary operator, a fraction is a single term.

Do you want to tell me that fractions don't take two numbers? That two numbers applied to division don't form a term?

Multiplication is also a binary operator, a and division is the opposite of it

Inverse. I read elsewhere that you're a math teacher and this is just such a perfect example of what's wrong with math ed: Teachers don't even know the fucking terminology. You don't know maths. You know a couple of procedural rules you shove into kids, rules that have to be un-taught in university because nothing of it has anything to do with actual maths.

No, it’s a mnemonic to remind people of the actual rules.

There's no such thing anywhere but in the US. Those rules are a figment of the imagination of the US education system.

You are up to your scalp in the Dunning-Kruger effect. Two possibilities: You quadruple down and become increasingly bitter, or you find yourself an authority that you trust, e.g. a university professor, and ask them in person. Ask a Fields Medalist if you can get hold of one. You think you know more about this than me. Motherfucker you do not, but I also acknowledge that I'm just some random guy on the internet to you.

If you want to continue this, I have one condition: Explain, in your own words, the difference between syntax and semantics. If you have done that, done that homework, I'm willing to resume your education. Otherwise, take the given advice and get lost I've got better things to do than to argue with puerile windbags.

[–] [email protected] 1 points 9 months ago (1 children)

That two numbers applied to division don’t form a term?

Now you're getting it! Correct, they don't. They form an expression. Terms are separated by operators, and joined by grouping symbols. Expressions are made up of terms and operators (since, you know, operators separate terms). I told you way back in the beginning that 1÷2 is 2 terms, and ½ is 1 term. Getting back to the original question, 2(2+2) is 1 term and 2x(2+2) is 2 terms.

you find yourself an authority that you trust

Which time that I mentioned textbooks, historical Maths documents, and proofs did you miss?

university professor

University professors don't teach order of operations - high school teachers do. That's like saying "Ask the English teacher about Maths".

If you want to continue this

Why would I want to when you ignore Maths textbooks and proofs? See my first comment in this post that you've finally got the difference now. See ya.

[–] [email protected] 2 points 9 months ago

They form an expression. Terms are separated by operators, and joined by grouping symbols. Expressions are made up of terms and operators (since, you know, operators separate terms).

Terms, expressions, symbols, all those are terms about syntax. Not semantics. Do you start to notice something?

Why would I want to when you ignore Maths textbooks and proofs?

To learn. I challenge you again to explain the difference between syntax and semantics. Last chance.