I guess I'm a real fan and I just didn't know it, because I check every day for uploads but I didn't recognize the inequality as specifying the floor function.
I think I was able to guess from the statement alone. The inequality makes it look like the function is sort of linear (with an error of about 1 unit at most), and it also specified that f(x)
Question: If you change the values in the orange condition, what (family of) function(s) do you get? If you remove the green condition, what family of functions do you get? If you remove the former blue inequality, what family of functions do you get? If you remove the latter blue inequality what family of functions do you get? Are any of these questions realistically answerable?
I asked myself the same questions and have some partial answers. There are other solutions then. If, for instance, you drop the red condition, you can multiply by any element a
I never really understood why the floor function was his favourite. But now he's converted me to a floor function enjoyer, functional equations always delight me and this was a particularly beautiful one!
technically he only proved that if a function with these properties exist, then it must be the floor function. however it is rather easy to see that the floor function indeed satisfies the stated conditions
I thought it was wrong at first, too. I had to analyze it a bit to understand. Lol I think swapping "f(x) + f(y) + 1" and "f(x) + f(y)" and flipping the inequalities would make it less confusing.
I think the extra two assumptions are too specific to feel like it's some natural thing, most wspeciall the second one. If you assume f is increasing, then saying [0,1) all
Just a variation of the & symbol, used within linguistics too in some cases. Which one to use is usually just up to the preference of whoever's writing it.
As is said below, that's just a graphical variant of the ampersand symbol. The ampersand symbol is just a digraph (read: combination that retains the original order) of "et", the Latin word for "and". That variant symbol is really supposed to be an E with the two lines above and below representing the t, but, for some unknown reason Michael wrote the symbol backward. (Yes, they aren't really dots, but small line segments, kind of like how some people don't put a full line through the S when writing a $, but just put little lines on the top and bottom of the symbol.). There's another variant that's sometimes used that is literally just a curvy Et, with the bottoms joined together (It helps if you use the lowercase t with the serif on the bottom, like in the font UA-cam uses, but flipped the other way around.). If you write that and kind of tilt the top of the T leftward a little, then you'll get an ampersand. Also, I'd like to point out that the name "ampersand" comes from two facts: one is that the ampersand used to be considered an actual letter of the alphabet in English, and the second is the Alphabet Song. In the song, the last letter was the ampersand, but it was originally called "per se and". "Per se" is Latin for "for itself", so "per se and" meant the symbol was the word "and" in its own right. Since "per se and" was the last letter of the alphabet, the song ended with the words "and per se and", which eventually was re-bracketed and shortened to "ampersand".
You know you're a real fan when you immediately realise the function is the floor function only from the video's title.
Nooo! You spoiled me the video.
I guess I'm a real fan and I just didn't know it, because I check every day for uploads but I didn't recognize the inequality as specifying the floor function.
why is this the top comment, fucking spoilers
@@9WEAVER9It's not about the inequality, but rather fhe video's title.
I think I was able to guess from the statement alone. The inequality makes it look like the function is sort of linear (with an error of about 1 unit at most), and it also specified that f(x)
The thumbnail is a bit confusing
Definetly
Oh, thank God! The thumbnail had me questioning all of reality.
I always like functional problems
Yesa, I like them, too. But I am not good at solving most of the time.
@@hassanalihusseini1717 they are fun
16:58 I know the answer and I won’t give it, but yeah that’s a classic from Michael 😂
Question: If you change the values in the orange condition, what (family of) function(s) do you get? If you remove the green condition, what family of functions do you get? If you remove the former blue inequality, what family of functions do you get? If you remove the latter blue inequality what family of functions do you get? Are any of these questions realistically answerable?
I asked myself the same questions and have some partial answers. There are other solutions then. If, for instance, you drop the red condition, you can multiply by any element a
try it yourself
So there is just 1 function that satisfies the criterias and it is the floor function?
y=x also works, right?
@@lyrimetacurl0not quite... There's the not-in-thumbnail inequality 2
I never really understood why the floor function was his favourite. But now he's converted me to a floor function enjoyer, functional equations always delight me and this was a particularly beautiful one!
Cool proof and amazing presentation. Thanks!
Did we prove that this is the only function that satisfies this property?
Yes.
technically he only proved that if a function with these properties exist, then it must be the floor function. however it is rather easy to see that the floor function indeed satisfies the stated conditions
I belive the cover of the video is wrong
I thought it was wrong at first, too. I had to analyze it a bit to understand. Lol
I think swapping "f(x) + f(y) + 1" and "f(x) + f(y)" and flipping the inequalities would make it less confusing.
It's not
@@assassin01620 it is saying that f(x)+f(y)+1
@@lucianoxiccato5458 The red arrows are telling you where to place the "f(x) + f(y) + 1" and the "f(x) + f(y)". So it's saying:
"f(x) + f(y)
@@assassin01620 AHHHHH sorry but that was confusiong ahah
Loved the use of its period at the end of the proof!
I started by substituting 0 for y. I didn't substitute anything specific for x in this
The floor here seems to be made out of floor
Not a big deal, but the card has the greater thans flipped?
you're missing the red arrows
@@MrStanny32 indeed I did. I think reversing the three lines would be clearer still.
Cool video, but dont you still have to proove that floor(x) fulfills those conditions?
I think the extra two assumptions are too specific to feel like it's some natural thing, most wspeciall the second one. If you assume f is increasing, then saying [0,1) all
Congratulations sir for 300k subscribers . You are my one of the favourite maths teacher ❤
Interesting, amene, and very well explained. Thanks❤
Biggest ever clickbait on the cover
Just one glance tells me it's the floor function (let's see the proof and whether I'm right)
More new stuff!
OH NO. I SMELL FLOOR. (Paused at 11:19)
You didn't need to prove f(x) = -1 for x in (-1,0) I think?
Why does f(x+1)=f(x)+1 represent the floor function
It doesn't. But if you know the values from 0 to 1 (which he calculated earlier) you can use this formula to calculate all of the others.
What symbol are you using instead of &? It looks like the number 3 with a dot above and below. Is this something people are generally familiar with?
Just a variation of the & symbol, used within linguistics too in some cases. Which one to use is usually just up to the preference of whoever's writing it.
As is said below, that's just a graphical variant of the ampersand symbol. The ampersand symbol is just a digraph (read: combination that retains the original order) of "et", the Latin word for "and". That variant symbol is really supposed to be an E with the two lines above and below representing the t, but, for some unknown reason Michael wrote the symbol backward. (Yes, they aren't really dots, but small line segments, kind of like how some people don't put a full line through the S when writing a $, but just put little lines on the top and bottom of the symbol.). There's another variant that's sometimes used that is literally just a curvy Et, with the bottoms joined together (It helps if you use the lowercase t with the serif on the bottom, like in the font UA-cam uses, but flipped the other way around.). If you write that and kind of tilt the top of the T leftward a little, then you'll get an ampersand.
Also, I'd like to point out that the name "ampersand" comes from two facts: one is that the ampersand used to be considered an actual letter of the alphabet in English, and the second is the Alphabet Song. In the song, the last letter was the ampersand, but it was originally called "per se and". "Per se" is Latin for "for itself", so "per se and" meant the symbol was the word "and" in its own right. Since "per se and" was the last letter of the alphabet, the song ended with the words "and per se and", which eventually was re-bracketed and shortened to "ampersand".
12:58 I know the answer now 😃
Yay
Watch complete video 🎉
thumbnail is wrong way around?
Why would this be someone's favorite function. It seems really boring. Please explain why interesting.
That is wring.every function is special in themselves.
The title is a Spoiler
The inequality immediately tell you that atleast the floor function is one of the solutions
whoa sometimes I think I should watch videos about chess problems. where did that --- nvm keep goin'
Very cool clickbate intro video image. I thought it was saying "f(x)+f(y)+1
the conditions are just so obviously tailored to the specific function in question that the problem becomes rather boring
Too hard.
Downvoted. From the thumbnail there are a whole class of solutions, thus it doesn't "define" any function, let alone your favorite.