Make sure to also throw in some ∇, as well as Δ’s doppelgänger Δ, the primes (‘, ‘’, ‘’’,…), their shorthand superscripts, the dots, and the subscripts.
@@quertyVAT 29/40, which is much better than I thought I'd score. Because correlation *does* mean causation, I can only assume my success was can be attributed to Zundamon. Thank you Zundamon.
Is there an argument explaining why they are equal? We don’t know that having the property where your forward difference returns the original function is a unique property. I see that numerically for small examples they’re the same but curious what the proof would be.
@@Happy_Abe There is. Since you replace the power x^k by the decreasing product x(x-1)(..)(x-k+1), you recognise x(x-1)(..)/k! as the (generalized) binomial coefficient C(k in x), and the sum for all k is a well known result = 2^x, that can be proved by writing a binomial series (1+z)^x = Σ C(x k) z^k, with z = 1.
6:07 The "formal name" of this is the "falling factorial". Also, reversely, there is a "rising factorial", which is used to define hypergeometric functions.
@@catmacopter8545 Pochhammer symbol. Personally, this symbol makes me quite a bit confused, so I would rather use Donald Knuth's notation (introduced in this video) which is more intuitive to me.
I love when TH-cam realizes I love both maths and anime girls. This is an amazing channel dude, I can’t wait to see more of it. This will be helpful to teach my friends some calculus when they eventually struggle with it lol
It almost seems like this should be taught before calculus and then you move to limits in calculus and see the beautiful applications of jumping to the continuous world
Me, way back when I was playing Earthbound, and I noticed the EXP required to reach the next level was interesting. I had taken calculus, and used differences and was somehow able to reason through it to find the polynomial for the level/EXP required to reach the next level correspondences for up to Level 18 or so, when the polynomial changed. It was a cubic polynomial. My math teacher told me this was finite differences. Since then, I’ve found the subject fascinating.
Thank you Zundamon and Metan. I like your videos very much. I don't understand all of them, but I hope that one day I will understand and enjoy them to the fullest.
considering that "e" has to do with continuous compounding interest at 100%, the connection to "2" is obvious, since 2^x is just 100% compounding interest at compounding time of 1.
It's funny how I, and some others in a discord server I'm in, were working on this at roughly the exact same time this video was released. We got analogues of a lot of things, and I managed to get the antidifferentiation at the end to work for complex values of h (or Delta), albeit with a lot of jank.
The funny part of this is that I feel like I learnt a lot of these concepts way back in Year 7 - I think it may well have been the first maths topic I was taught at comprehensive school - but it was under the topic heading of "nth term" rather than talking about discrete calculus. And then it never really came up again, even once I started doing calculus at A-level, until eventually the idea of "difference equations" (as opposed to differential) came up. Feels like the idea could have been developed more at comprehensive school, and then used as a bridge to regular continuous calculus.
That was fun! I mean, going from continuous to discrete numbers is fun! My hypothesis on why 2 is similar to e when going into the differences, is that 2 is the integer part of e, and so when you move from continuous to discrete you mostly take the integer part of numbers and work with them. Let me know if my theory makes sense 😅
That's a nice intuition, but the real key fact is that if you take the series formula for e^x and replace each power x^n with a "falling power"/"falling factorial", then the series evaluates to 2^x.
It seems like if you tried to find an antiderivative in this world, assuming your step value is h, the result would not be the same as in regular calculus, where the antiderivatives differ only be a constant. This would be the case (the antiderivatives differing by a +C) if our function only takes on values of the form k + n*h, where k is some constant real number and n is every integer. This would also be the case if we imposed the restriction that the antiderivative be continuous, however not all functions would have a continuous antiderivative in this world even if they have an antiderivative. For example, the sign function with f(0) = 1 has derivative 1 on [-h,0) and 0 elsewhere. This derivative cannot possibly have a continuous antiderivative. Assume this derivative has a continuous antiderivative g. Observe that g must have the following properties: for all 0.5 > ε > 0, there exists a δ > 0, such that 0 < | x - a | < δ implies | g(x) - g(a) | < ε, and for all h > δ_0 > 0, g( h - δ_0) = g( -δ_0 ) + 1, and g(h) = g(0). This derives from the definition of continuity and the derivative of g. Fix 0.5 > ε > 0 and consider continuity at the points x = 0 and x = h. There exist 0 < δ_1, δ_2 < 0.5, such that | x - 0 | < δ_1 implies | g(x) - g(0) | < ε, and | y - h | < δ_2 implies | g(y) - g(h) | < ε. Let x = -δ_0 and y = h - δ_0 with δ_0 < δ_1, δ_2. Then, | g(-δ_0) - g(0) | < ε, and | g(h-δ_0) - g(h) | < ε. Using our properties, we get | g(-δ_0) + 1 - g(0) | < ε -ε < g(-δ_0) - g(0) + 1 < ε -1-ε < g(-δ_0) - g(0) < -1+ε g(-δ_0) - g(0) < -1+ε < -ε < g(-δ_0) - g(0), a contradiction. Therefore there cannot be a continuous antiderivative of the derivative of this sign function. Let me know if I can simplify this proof or if there is a simpler counterexample. This was just the first that came to mind. This shows that not all well defined functions can have any continuous antiderivatives, even piecewise smooth functions. Antiderivatives and derivatives in this world have a lot of different properties. In this world, almost all antiderivatives of continuous functions are continuous nowhere, and relatedly, all well defined functions have a derivative. All well defined functions have antiderivatives, but there exist piecewise smooth functions without continuous antiderivatives. In normal calculus, the antiderivative of a continuous function is continuous, but not all well defined functions have an antiderivative at all, since for example, there is no differentiable function whose derivative is discontinuous everywhere. Without further restrictions, the only thing we can say is that the antiderivative differs only by a +C from other antiderivatives at every set of points of the form k + n*h. I know nothing about this topic and am assuming people who actually work with this subject do impose some restrictions on the functions typically, but this is what I conclude from what the video presents. Corrections welcome
I suppose what I said at the end there about the existence of antiderivatives is slightly wrong if you consider the antiderivative as an integral, rather than a function that has f as its derivative?
10:43, I have an very good example to extend this to n-Dimensional space for general function, would like to make a video on that? From there we can actually show exterior calculus with tensors of various forms, which in topology very much helps to separate line to area vectors. Let me know if we have a chance to collaborate on this
one for L hopital rule video plz i am a jee aspirants and tbh i need this my exam is in 40 days i loved it i am going to bing watch your every video tonight
this is the forward difference - there is also the backwards difference f(x)-f(x-1) = ∆f(x-1). i recently read about the "symmetric derivative", whos analogue here would be (f(x+1)-f(x-1))/2. Does that have any interesting properties (besides for being the average of the next and previous elements of the sequence)
@@wargreymon2024 yeah just as with calculus the difficult part comes when solving sums one of the mid-level exercises on summation by parts in Concrete Maths escaped me the last few times I tried it
thanks a lot.. i always wondered what could be the analog of calculus if we discretise numbers. I got this doubt while thinking whether anyone tried using discrete math so it might fit better for quantum mechanics ( i just thought so, i didn’t read anywhere)
Please check the video description! 😆 As for the voices, they need to be combined with other tools. You can also find videos trying the same challenge on TH-cam! 👍
Δ,d and ∂ would make good characters in a saturday morning cartoon.
D and δ also
when animation vs math (alan becker) goes into calculus, maybe the gang can educate the orange stick figure
(and the integral symbol looms around)
@@JR13751Wow,there are lots of D's in math
Math cartoons are a good idea anyway!
Make sure to also throw in some ∇, as well as Δ’s doppelgänger Δ, the primes (‘, ‘’, ‘’’,…), their shorthand superscripts, the dots, and the subscripts.
"No, I'm just reading it out loud" I'm dying
Supposed to be studying for a test that's got nothing to do with discrete calculus at the moment, but a new Zundamon's Theorem takes precedence.
same here
what did you score
@@quertyVAT 29/40, which is much better than I thought I'd score.
Because correlation *does* mean causation, I can only assume my success was can be attributed to Zundamon. Thank you Zundamon.
9:42 It's deeper than that. If you define the exponential as e^x = Σ x^k / k! and replace the regular power by the falling power, you get exactly 2^x.
Is there an argument explaining why they are equal?
We don’t know that having the property where your forward difference returns the original function is a unique property. I see that numerically for small examples they’re the same but curious what the proof would be.
@@Happy_Abe There is. Since you replace the power x^k by the decreasing product x(x-1)(..)(x-k+1),
you recognise x(x-1)(..)/k! as the (generalized) binomial coefficient C(k in x), and the sum for all k is a well known result = 2^x, that can be proved by writing a binomial series (1+z)^x = Σ C(x k) z^k, with z = 1.
@@Risu0chan wow that’s great thanks a lot makes a lot of sense. Awesome result!
6:07 The "formal name" of this is the "falling factorial". Also, reversely, there is a "rising factorial", which is used to define hypergeometric functions.
I was gonna comment that
it's also very related to the Pocchammer symbol or however it's spelled
@@catmacopter8545 Pochhammer symbol. Personally, this symbol makes me quite a bit confused, so I would rather use Donald Knuth's notation (introduced in this video) which is more intuitive to me.
now we are learning backwards
We're literally forgetting
well, if forwards thinking doesn't work, gotta try backwards, right? :D
@@simdimdim Go back to monke
I love this and the jp channel so much like i actually cried because of how cute it is
3:11 LITERALLY ("literally" not being sementically bleached, I love it.)
"SEMENtically"
Lmaoooooo😂😂😂😂
@@Sir_Isaac_Newton_ haha you sure are a Fungi
10:38 "could it be?" SOOOOO CUTE
I love when TH-cam realizes I love both maths and anime girls.
This is an amazing channel dude, I can’t wait to see more of it. This will be helpful to teach my friends some calculus when they eventually struggle with it lol
thank you as always zundamon’s theorem en for the absolute cinema
I like the silly parts of dialogue in this, it makes them feel more like people
This channel has a huge potential to grow!
I love refreshing my understanding of calculus and learning new things, thanks for this experience.
oh of course, as integration is a continuous sum, the sum is the discretization of the integral
Thank you. This opened my mind to a realization never before considered
@@JactheConsumergood for you
I love you Zundamon's Theorem.
"If you already know the definition of differentiation, please skip to 1:16" Who are you talking to, Metan? 😆
She broke the fourth wall, I believe!
And the fact that the diference of polynomials is so much thougher than their diferentiation is a reason why it's so much thougher
It almost seems like this should be taught before calculus and then you move to limits in calculus and see the beautiful applications of jumping to the continuous world
6:08 me whenever I try to flex in front of my friends:
Weak lmao
@@Johnny-tw5pr cant argue lol
Me, way back when I was playing Earthbound, and I noticed the EXP required to reach the next level was interesting. I had taken calculus, and used differences and was somehow able to reason through it to find the polynomial for the level/EXP required to reach the next level correspondences for up to Level 18 or so, when the polynomial changed. It was a cubic polynomial. My math teacher told me this was finite differences. Since then, I’ve found the subject fascinating.
This is such an amazing format to teach math in. Thank you for sharing.
Thank you Zundamon and Metan. I like your videos very much. I don't understand all of them, but I hope that one day I will understand and enjoy them to the fullest.
I love zundamon's videos a lot
Ooh, the summation corresponding to integration is very interesting. Thank y'all for another banger video.
This video explained derivative much much way better than my college teacher! Outstanding!!!!
Bro, your videos are really good, i dindt even knew about the existence of discrete differentiation, as a calculus student i appreciate your work👍👍👍
Zundamon , can you do a video on partial differentitation?
Ya much needed
I love these videos, they give me good perspectives into maths as a whole.
7:16 you explained the gamma functions so easily, nice.
The peak is back
Beautiful Beautiful Beautiful. one day this channel will reach millions this is great content. Thanks alot❤❤❤
I'm watching the World Chess Championship 2024, but a new Zundamon's Theorem video takes priority.
Maths, a world of limitless possibilities, (until you hear about Gödel and his work)
"EN Zundamon isnt real, it cant hurt you."
EN Zundamon:
considering that "e" has to do with continuous compounding interest at 100%, the connection to "2" is obvious, since 2^x is just 100% compounding interest at compounding time of 1.
great video as always ! I wanna be a math youtuber like u 2 one day
funky stuff, now i know about differentiation, partial differentiation, and differences. math is funky
It's funny how I, and some others in a discord server I'm in, were working on this at roughly the exact same time this video was released. We got analogues of a lot of things, and I managed to get the antidifferentiation at the end to work for complex values of h (or Delta), albeit with a lot of jank.
Love these vids! Laughed good when I heard:
"let's leave the exercise to our readers..."
"You mean viewers" 😂
The funny part of this is that I feel like I learnt a lot of these concepts way back in Year 7 - I think it may well have been the first maths topic I was taught at comprehensive school - but it was under the topic heading of "nth term" rather than talking about discrete calculus. And then it never really came up again, even once I started doing calculus at A-level, until eventually the idea of "difference equations" (as opposed to differential) came up. Feels like the idea could have been developed more at comprehensive school, and then used as a bridge to regular continuous calculus.
Lil bro open a page for people to donate / support your work this stuff is so unique and fun and cool and interesting
i recall mathloɡer doinɡ a similar thinɡ
English Zundamon teaching math? This is what technology was made to do.
I always find videos like this so amusing
Anime girl math is all I needed in my life right now and I didn't even know it.
wow i randomly thought of this while trying to find the area under the graph of a sequence lol. didnt know it was called discrete calculus, thanks
Zundamon if you are taking suggestions it would be really really cool if you looked into the Fast Growing Hierarchy
Never thought about how these might relate before. Interesting find
Just so you know this brach of math is called umbral calculus
Zundamon, please share the references for your videos for people who want to delve deeper into the topics!
why is this channel so good? its not even just the waifus making them this good thats just a bonus
10:43 the way my jaw dropped
Another excellent video thank you!
this guys is legit gonna be famous in a year
It's like Calculus but you turned on easy mode. Love it🥰
Thank you zundamon!
That was fun! I mean, going from continuous to discrete numbers is fun!
My hypothesis on why 2 is similar to e when going into the differences, is that 2 is the integer part of e, and so when you move from continuous to discrete you mostly take the integer part of numbers and work with them.
Let me know if my theory makes sense 😅
That's a nice intuition, but the real key fact is that if you take the series formula for e^x and replace each power x^n with a "falling power"/"falling factorial", then the series evaluates to 2^x.
@@diribigal Interesting!
You've obtained something like differentiation
It seems like if you tried to find an antiderivative in this world, assuming your step value is h, the result would not be the same as in regular calculus, where the antiderivatives differ only be a constant.
This would be the case (the antiderivatives differing by a +C) if our function only takes on values of the form k + n*h, where k is some constant real number and n is every integer.
This would also be the case if we imposed the restriction that the antiderivative be continuous, however not all functions would have a continuous antiderivative in this world even if they have an antiderivative. For example, the sign function with f(0) = 1 has derivative 1 on [-h,0) and 0 elsewhere. This derivative cannot possibly have a continuous antiderivative.
Assume this derivative has a continuous antiderivative g. Observe that g must have the following properties: for all 0.5 > ε > 0, there exists a δ > 0, such that 0 < | x - a | < δ implies | g(x) - g(a) | < ε, and for all h > δ_0 > 0, g( h - δ_0) = g( -δ_0 ) + 1, and g(h) = g(0). This derives from the definition of continuity and the derivative of g.
Fix 0.5 > ε > 0 and consider continuity at the points x = 0 and x = h. There exist 0 < δ_1, δ_2 < 0.5, such that | x - 0 | < δ_1 implies | g(x) - g(0) | < ε, and | y - h | < δ_2 implies | g(y) - g(h) | < ε. Let x = -δ_0 and y = h - δ_0 with δ_0 < δ_1, δ_2. Then, | g(-δ_0) - g(0) | < ε, and | g(h-δ_0) - g(h) | < ε. Using our properties, we get
| g(-δ_0) + 1 - g(0) | < ε
-ε < g(-δ_0) - g(0) + 1 < ε
-1-ε < g(-δ_0) - g(0) < -1+ε
g(-δ_0) - g(0) < -1+ε < -ε < g(-δ_0) - g(0),
a contradiction. Therefore there cannot be a continuous antiderivative of the derivative of this sign function. Let me know if I can simplify this proof or if there is a simpler counterexample. This was just the first that came to mind. This shows that not all well defined functions can have any continuous antiderivatives, even piecewise smooth functions.
Antiderivatives and derivatives in this world have a lot of different properties. In this world, almost all antiderivatives of continuous functions are continuous nowhere, and relatedly, all well defined functions have a derivative. All well defined functions have antiderivatives, but there exist piecewise smooth functions without continuous antiderivatives. In normal calculus, the antiderivative of a continuous function is continuous, but not all well defined functions have an antiderivative at all, since for example, there is no differentiable function whose derivative is discontinuous everywhere. Without further restrictions, the only thing we can say is that the antiderivative differs only by a +C from other antiderivatives at every set of points of the form k + n*h.
I know nothing about this topic and am assuming people who actually work with this subject do impose some restrictions on the functions typically, but this is what I conclude from what the video presents. Corrections welcome
i shouldve typed this is LaTeX smh
I suppose what I said at the end there about the existence of antiderivatives is slightly wrong if you consider the antiderivative as an integral, rather than a function that has f as its derivative?
Calc 1 Students Be Like:
No Way We Goin Through That Hell Again
umbral too
Zundamon Is AN art
Great video as always ❤❤
amazing channel
Umbrella calculus Is Better then raining
Estou aprendendo Cálculo por agora e o canal está me ajudando e muito!! Obrigado❤❤
10:43, I have an very good example to extend this to n-Dimensional space for general function, would like to make a video on that? From there we can actually show exterior calculus with tensors of various forms, which in topology very much helps to separate line to area vectors. Let me know if we have a chance to collaborate on this
dewivative
this... this is
Math at its finest
10:35 I literally pogged this shit is so poggers wowee
dewivatiwe :3
i have been binging on your videos and i loved every one of them! do you have a patreon or some other platform to support this channel?
that's just learning derivatives backwards
umm... its more than a week now
one for L hopital rule video plz i am a jee aspirants and tbh i need this my exam is in 40 days i loved it i am going to bing watch your every video tonight
this is the forward difference - there is also the backwards difference f(x)-f(x-1) = ∆f(x-1).
i recently read about the "symmetric derivative", whos analogue here would be (f(x+1)-f(x-1))/2. Does that have any interesting properties (besides for being the average of the next and previous elements of the sequence)
AAH, it’s like that, huh. I understand everything now
(Doesn’t get it all)
it's an introduction, the rabbit hole is deep on this topic...
@@wargreymon2024 yeah just as with calculus the difficult part comes when solving sums
one of the mid-level exercises on summation by parts in Concrete Maths escaped me the last few times I tried it
This is so damn good
Do P-adic numbers next!!!!
I need help with those ngl
Zundamon, could you fix your profile picture? It would be nice if Zundamon looked to the right. This way you can leave space for the heart.
where in math is this used? i've never seen such thing!
the conexions to normal calculus are so interesting
thanks a lot.. i always wondered what could be the analog of calculus if we discretise numbers. I got this doubt while thinking whether anyone tried using discrete math so it might fit better for quantum mechanics ( i just thought so, i didn’t read anywhere)
Now... Is there a discrete analogue of the Euler identity?
Wonderful. I will be playing around with integration.
ok, wolfram alpha told me that Δ(Polygama(0,x)+c)=1/x?????
Cool video as always. I learned something new. But now i wonder, what would be the derivative of a falling power, I will check it.
8:08 Pierre de Fermat reference
This is just touching the surface. The techniques for solving difference equations closely mirrors the techniques for solving differential equations.
The style reminds me of yukkuri from touhou
Fascinating
Pls do statistics 🙏
Oh what?! These are in English now??
This is amazing! Tell me, how do they speak English? Voicevox only understands Japanese((
It needs to be combined with other tools.
You can find videos attempting the same challenge on TH-cam! 👍
What about a continuous product?
I'm curious, did you draw the characters yourself? Also, I would love to learn more about what you are using for the character voices.
Please check the video description! 😆
As for the voices, they need to be combined with other tools.
You can also find videos trying the same challenge on TH-cam! 👍
Zooba! 0w0 cant wait for linear algebra
Yay another anime girls math video
I Copy summary and explain myself every point.
I saw this in physics
PHYSICS!???!!?!?!!!?
@@z0ru4_ yes!
@@z0ru4_Theoretical physics: 👁👄👁
uhh what the sigma