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.
I'm not British. So you're saying Maths doesn't work the way that Maths textbooks teach it - do go on...
Also Google...
2 (4) is the same thing as 2 * 4.
No, as I already pointed out, it's the same as (2x4). You can't remove brackets unless there is only 1 term left inside. 2x4 is 2 terms, so can't remove brackets yet.
Division and multiplication are equal in the order of operations
I didn't say they weren't. I said...
Doing division before brackets goes against the order of operations rules
You did 8/2x4, which is the same as (8/2)(2+2), which isn't the same as 8/2(2+2)=8/2(4)=8/(2x4).
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.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
They didn't - neither of them is a rule of Maths.
That you’re still wrong?
About? You haven't pointed out anything that's wrong.
the problem is written poorly due to the obelus and thus is open to interpretation
Oh, you're one of those people. Good, maybe we can finally get an answer then (this was also talked about in the blog). What other interpretation of an obelus is possible other than division? People keep saying it's ambiguous, but no-one has ever said why (other than some stuff that makes no sense in the context, as explained in the blog)
The distributive property is sometimes called the distributive law of multiplication and division
Yes, and sometimes people call Koalas "Koala bears", but that doesn't mean they're bears. Now bearing that in mind, read again what Khan said - the page which is called "Distributive property explained", not "Distributive Law explained".
Wait till you hear that “i before e except after c” wasn’t true either
Wait till you hear that's not a rule of Maths.
It’s wild that you think 7th grade math overrules grad school math though
Umm, never said anything of the kind...
Don't need any extra letters - just need people to remember the rules around expanding brackets in the first place.
there’s absolutely no difference in n(n-1) and n*(n-1)
There is - the first is 1 term and the 2nd is 2 terms. Makes a difference if it's preceded by a division.
it’s just matter of convenience you can leave it off.
It's a matter of how many terms as to whether it's there or not.
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.
Which time that I mentioned textbooks, historical Maths documents, and proofs did you miss?
University professors don't teach order of operations - high school teachers do. That's like saying "Ask the English teacher about Maths".
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.