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.
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?
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).
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.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?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).
It’s called Terms - Terms are separated by operators and joined by grouping symbols. i.e. ab=(axb).