Why don't they teach Newton's calculus of 'What comes next?'
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- čas přidán 13. 06. 2024
- Another long one. Obviously not for the faint of heart :) Anyway, this one is about the beautiful discrete counterpart of calculus, the calculus of sequences or the calculus of differences. Pretty much like in Alice's Wonderland things are strangely familiar and yet very different in this alternate reality calculus.
Featuring the Newton-Gregory interpolation formula, a powerful what comes next oracle, and some very off-the-beaten track spottings of some all-time favourites such as the Fibonacci sequence, Pascal's triangle and Maclaurin series.
00:00 Intro
05:16 Derivative = difference
08:37 What's the difference
16:03 The Master formula
18:19 What's next is silly
22:05 Gregory Newton works for everything
28:15 Integral = Sum
32:52 Differential equation = Difference equation
36:06 Summary and real world application
39:22 Proof
Here is a very nice write-up by David Gleich with a particular focus on the use of falling powers. tinyurl.com/ymcyrapz
This is a nice lesson from a Coursera course on this topic www.coursera.org/lecture/disc...
One volume of Schaum's outlines is dedicated to "The calculus of finite differences and difference equations" (by Murray R. Spiegel) Examples galore!
This is a really nice very old book Calculus Of Finite Differences by George Boole (published in 1860!) tinyurl.com/3bdjr932 Thanks to Ian Robertson for recommending this one.
There is a wiki page about our mystery sequence: tinyurl.com/uwc89yub It's got a proof for why the mystery sequence counts the maximal numbers of regions cut by those cutting lines. If you have access to the book "The book of Numbers" by John Conway and Richard Guy, it's got the best proof I am aware of.
Here is a sketch of how you solve the Fibonacci difference equation to find Binet's formula imgur.com/a/Btu5ZVk
Here are a couple more beautiful gems that I did not get around to mentioning:
1. When we evaluate the G-N formula for 2^n what we are really doing is adding the entries in the nth row of Pascal's triangle (which starts with a 0th row :) And, of course, adding these entries really gives 2^n.
2. Evaluating the G-F formula for 2^n at n= -1 gives 1-1+1-1+... which diverges but whose Cesaro sum is 2^(-1)=1/2!! Something similar happens for n=-2.
3. In the proof at the end we also show that the difference of n choose m is n choose m-1. This implies immediately that the difference of the mth falling power is m times the difference of the m-1st falling power.
Today's music is by "I promise" by Ian Post.
Enjoy!
Burkard
P.S.: Some typos and bloopers
• Why don't they teach N... (396 should be 369)
• Why don't they teach N... (where did the 5 go?)
• Why don't they teach N... (a new kind of math(s) :)
Back with another crazy long one. Hope you like it :)
Always😀😀
💙💚🙏🤲🫂😅
U make video about at the rate of 1 per month. But that whole 30-50 minutes video made my day, seriously. Lots of love ❤❤❤ prof from India🇮🇳. But I would love if u increase the rate..
Ofc we will!
Yes sir
Every part of calculus is mirrored in sequence calculus EXCEPT the chain rule. This is what makes infinitesimal calculus extraordinarily more powerful.
Really? Well maybe one day we will find a way to include the chain rule that helps us see the difference more clearly.
Maybe one day, it will make sense.
Of coutse, this video is already perfect, but I can't wait to see what the mathematics community has next!
You can't even form the composition unless one of the sequences is integer-valued. If it is, the chain rule remains the same.
Sounds like you just casted a spell.
@@calculusillustrated2854 I think it is also possible for rational values
@@calculusillustrated2854 The chain rule never remains the same, even if the sequences are integer valued.
9:39 I love this thing about 2^n being equivalent to e^x. It's as if someone came along and very simplistically assumed that "we're dealing with integers, so we should round e down to 2". It's crazy! :D
You should check out the Cauchy condensation test, which is used to evaluate the convergence of infinite series. It draws on the similarities of 2^n in sequence calculus with e^x in differential and integral calculus while also relating the discrete sum with the continuous sum (integral). Seeing this test really blew my mind after watching this video. Here's a link to it on Wikipedia: en.wikipedia.org/wiki/Cauchy_condensation_test
@@violintegral very cool! Thanks!
Lies again? Hello Whatsapp
Remember what "e" represents though - it's the compounded growth rate in infinitesimal time periods over a unit of time of something that would have doubled over that same unit of time if divided into only one period of growth.
Take an annual interest rate of 100% (i.e. doubling). Now divide that growth into smaller units and apply the growth over each of those smaller units instead of adding it only at the end - e.g. 100/365 % added per day leads to growth (1 + 1/365)^365 = 2.714.. In the limit we'd get "e", hence "e" is to continuous time what "2" is to discrete time.
Any exponential function may it be x^n behaves like e^x when quantized. Chicks double each day. so dx/dt != 2^n
but the difference between f(n) - f(n-1) is indeed 2^n.
Donald E. Knuth calls this "finite calculus", as opposed to the usual "infinite calculus" that is commonly taught. His book, "Concrete Mathematics: A Foundation for Computer Science", uses this "finite calc" to tackle subjects such as hypergeometric functions, generating functions and asymptotics, and derives lots of analogies between the two variants. For example, establishing a power rule, an analogue to exponentional and logarithmic functions and even a "summation by parts" technique. Finite calc is a powerful tool that lets us work wonders, and makes lots of sums we usually cant tackle, easily reducible. I'd say check the book out!
In my time it was called the calculus of finite differences. Love the Mathologer videos.
I did wonder if it would appear in any "discrete math" or other "math for computing" contexts but my more advanced algorithms texts (where it might be found) are elsewhere. It is not in the DM book I have here.
Concrete mathematics is also one of my favourite textbooks. I recently mentioned it in this video (something about generating functions) czcams.com/video/VLbePGBOVeg/video.html
Yeah the book is extremely cool to read! The cherry on top tho? All the funny commnetary in the book and written in its margins😁😁
The exercises in that book are tough. I remember being able to prove or solve very few.
I love this process. We learnt this before moving on to “conventional” calculus at my school and I took it for granted that everyone had too.
Would be a very natural thing to do but sadly hardly anybody ever gets to know about this beautiful topic (until now of course :)
I'm just realising we did too, but no emphasis was placed on it, nor was it's source explained
I didn't knew anything about it up to now.
Lol I stumbled into it because I on and off for a few years looked at varying degrees of triangular sums and their closed forms. Of course I didn't arrive anywhere from that other than just saying "oh well isn't this neat".
I eventually, looking at matrix algebra, blundered into the idea of using systems of equations to determine coefficients for the general polynomial solution which disappointingly contented my former self.
Faulhaber's formula is what you want to look at if you want all the closed form solutions for polynomial sums/series. It's interesting.
Was the calculus curriculum you described following a particular textbook? If so, what was the title and author of the textbook? All I remember about sequences is the very boring epsilon-delta proofs...
In 2013, I discovered this Gregory-Newton formula myself using insight from Pascal's triangle. I found the formula while trying to solve an arithmetic sequences and sums problem with 4th difference for a private high school student I taught.
I had no idea how to answer the question using the standard arithmetic sequence and sum formulas taught in schools. I struggled with that one question for more than an hour before I found an insight in Pascal's triangle and then came up with this helpful formula.
I tested the formula with couples of made up cases to make sure the formula works and indeed it worked! I then told my student to use this formula to solve that particular question.
The next day, my student told me the answer was correct, but his teacher didn't give him full mark because he didn't use the standard formula. 💔
I didn't know the name of the formula I had discovered until today. But at that time, I was sure someone else should have found the formula. Imagine the humiliation I would get if I had named that formula by my name. 😅
Thank you for this video, sir. This brings back all the memories of that moment. 🙏🏼
That's great :)
Let's call it the Gregory Newton Nidalapisme formula.
The Newton-Leibniz formula was discovered by both of them independently.
Funny enough I did this but with pythagorean triples.
Was watching videos about pythagorean theorems and I discovered a formula to find the next whole number hypotenuse in a sequence of triangles.
I’ve watched enough math on CZcams to immediately recognize that first sequence as the number of regions a circle is cut into when n+1 points on the circumference are connected. Without spoiling too much, I would highly recommend an old ThreeBlueOneBrown video titled “Circle Division Solution” to learn more about this problem and it’s brilliant solution.
Just watched it; that was an *excellent* follow-up, good recommendation!
Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes OEIS A000127
Thanks. This then gets into lie groups etc? Basically modeling a hyperconnected volume. Super relevant to cosmology
Lol me too
A, the old cakemorphic integers :)
Another masterpiece as always. I am a math nut, have been my whole life. When I go for a walk I think about numbers and sequences and formulas and physics. I spent my career as an engineer but now that I am at retirement I spend large amounts of time on 2 of my loves - math and physics. This is am absolutely fantastic channel. Thank you!
hope you have a great life onwards sir!
It's neat how this feels like a sort of "hacky" way of approaching calculus, if that makes sense. Colleges should teach this imo. Just from this 40 min video, I could see some of these concepts plugging into all sorts of annoying classwork problems we've done in calc and differentials
I think they do it in CS courses. There is a class called Discrete mathematics, and one of the subjects is Generating Functions. Not one-to-one with this presentation, but you 'formaly' derivate and integrate expansion series of 1/(1-x).
Agreed. We weren't exposed to anything quite this off-the-path either. A little bit of "find the difference" when it came to finding a pattern in a sequence, but nothing like what he demonstrated in this one, or that you can actually sway a value in as he did.
@@mattb2043I was a math major who took Discrete and, while I barely remember what was in the course besides pigeon holing, I do remember coming out with the overall feeling that I already knew all of what I was being taught. I feel like generating functions were in later Calc 1.
Wow… this whole finding differences of a set of points of a polynomial is something I figured out in 11th grade. I made this program on my calculator that you input a table of points and it will calculate the area under the curve, and I found as you say, for ‘well behaved nice functions’ it was super accurate. Kinda proud I was able to figure this all out with only knowing very basically calculus at that point in my life.
@Kelvin Wrong… by ‘know calculus’ I knew what a derivative was and integral was but didn’t really know how to solve them. And even then this is still very different on how to find area under the curve using only a set of points very accurately.
@Kelvin bru u riped their holes wide open
@Kelvin CZcams comments are not aimed at mathematicians, so the same level of precise semantics are not required. You are arguing over semantics.
To a normal person in this context, "discovered" equals "independently confirmed." "Solve" equals "evaluate." And while I completely understand obsession with precision, remember that its absence upsets you because YOU lack the ability to easily parse meaning from context, and neurotypical people do not require such levels of semantic precision.
@@phoenixshade3 Damn, you got Kelvin good there. This is one of the best and most utterly irrefutable roasts I've read in a youtube comment section.
@@nicolasguereca8337 bey blade rip it up
33:14 Marty does not like the Fibonacci sequence, and neither does Matt Parker. From this, we can deduce that having a first name that starts with "Ma" and contains a "t" predisposes you to disliking the Fibonacci sequence. Mathologer doesn't count because, presumably, that's not his actual first name.
Indeed, it’s Burkard.
Matt Parker prefers the Lucas sequence. Which is a bit silly, because the Lucas sequence is just the Fibonacci sequence wearing a thin disguise.
The real question is whether or not the Gregory Newton formula can be used to extend this pattern in any way...
@@PhilBagels So Matt Parker prefers the Lucas sequence because he considers is to be the "purest" (as far as I know, he's never used the word "purest" in this context, but that's not the point) of the family of sequences (as in the sequences where any term is the sum of the previous two terms, each one differentiated by its initial two terms).
Or maybe your theory only holds for names that begin with 'Ma' and not just have the letter t but have it in the 4th place... That would explain why Mathologer likes the Fibonacci sequence...
Can I have a Noble Prize already?
One of the coolest parts of the Gleich paper is that it leads very naturally to the question of how to express normal powers in terms of falling powers; e.g., n^3 = 1 + 7n + 6n(n-1) + n(n-1)(n-2). Equivalently, is there a pattern in the first numbers of the rows of the difference scheme for f(n) = n^k? Go read the paper in the description for the answer!
(n+1)^3 = 1 + 7n + 6n(n-1) + n(n-1)(n-2)
Isn't n^3 = n + 3n(n-1) + n(n-1)(n-2)?
This is cool, so I had to check it, and that expression is actually equal to (n+1)^3. Still cool, though. 😮
@@Bluhbear Oops good catch! I forgot I did f(0) = 1^3, f(1) = 2^3, ...
That's not original to Gleich. Others were doing all this stuff decades ago. It was collected in the book by Ronald Lewis Graham, Donald Ervin Knuth and Oran Patshnik: "Concrete Mathematics" (1988), which is well worth owning. I have the eighth edition, October 1992.
This was a foundation subject taught in actuarial studies long before the advent of PCs and spreadsheets. Much of the early work of actuaries relied upon such techniques to analyse empirical mortality and morbidity data in life insurance. It’s actually a cornerstone of a broader field called numerical analysis.
Another long forgotten subject that might merit your attention is spherical geometry, the basis for navigation and astronomical work. It’s parallels and extensions to Euclidean geometry are fascinating.
Yes spherical geometry is something I like a lot too. I even own a spherical blackboard :)
@@Mathologer Don't erase everything on it, or else it will become a black hole, and suck you in !
@@Mathologer I'm just picturing you stuck in a perfect sphere where the internal boundary is a blackboard.
x to doubt, actuaries are bad at math and worse at calculus - src am actuary (gladly left)
@@ravenecho2410 In fact some actuaries are fantastic at pure maths but most are what you would say “good at maths”. It’s prerequisite to study highest level at school to enter courses. As you’d know, but many wouldn’t, actuarial studies is a discipline of applied maths combining a diverse range of fields such as demography, finance, investment, commerce, marketing, accounting, risk analysis, modelling to problems in insurance, banking, health, pensions etc. So the emphasis is on applied maths to real world problem solving. Being brilliant at calculus or number theory is not of much use. if your friend chose another calling where such maths fields are useful, that’s fine. My son is an aeronautical engineer, who despite all the theoretical maths work at uni, discovered real world work generally didn’t require it.
The ideas of derivatives = differences, integrals = sums, and diff. eqs = difference equations were things I realized one by one during my senior year and graduate courses in university. I would have appreciated this a few years back, great video!
A teacher showed me this back in high school, and I thought it was really pretty. I had wanted to include it in my combinatorics series, but I had to cut it for time. Great to see it covered in Mathologer fashion.
I feel like this sort of "discrete calculus" was something I was vaguely aware might exist, but I've never seen it formalised like this before.
Same, I've been aware of it for quite a while, and even use it constantly to make certain estimations, but never really attempted to study it in a formal way. Well, I "use" it in the sense that just taking the rules from continuous calculus gives good enough approximations for discrete sequences for my purposes.
For example, it often comes in handy when estimating asymptotic time complexity of algorithms. Come across a sum of squares? It's O(n^3), because the integral of x^2 is x^3/3. A sum of 2^n is just O(2^n). And so on... It makes things so easy, and you don't even have to worry about the constants.
I used to doodle when I was little and make pyramids with numbers such as these. It has been like a recurring theme in my doodles. One time I tried to calculate the number of all possible unique trichords in a 12-note equally tempered octave (music theory), and this kind of thing popped up again out of nowhere. It keeps following me
What's better than an "AHA moment"? Over 40 minutes of endless "AHA moment"s, of course!
Loved the video, make more, Burkard! (And preferably faster :D )
Wow, damn, I sorta figured out a couple of these things for myself when I was in school (and later on in college) but I was never explicitly _taught_ any of it! It's so cool to see that my thoughts about "wait, is 2^x a kind of discrete version of e^x?" were actually a thing that people had studied and wasn't just a weird quirk!
I didn't get explicitly taught any of this until I learned about the Z-transform. I was asking myself even then why the hell they waited so long to teach something so useful.
Same. And it is why this was actually discovered first.
e = (1+1/infinity) ^ infinity
2 = (1+1)^1
that's how i understood it
It's weird to see you mentioning falling powers, and their derivative... a long time ago I was bored, and started writing down the math for such functions to pass the time. I came to it because of the combination formula! It's very cool to see it actually has a functional purpose, more than I thought! I definitely noticed it could be helpful, but I never thought it could be THIS. This is so basic, and yet so important for the rest of calculus we learn in school, it's really amazing. Teaching this first, at least a little, could really help with people's intuition of Calculus.
Maybe also have a quick look at the article by David Gleich that I link to in the comments. It focusses on a couple of uses of these falling powers :)
I never learned this at school or university, but now I'm in love! It seems like a way to teach/reach much richness of calculus without the (potentially) intimidating infinite limits and infinitesimals. I mean, we want to learn those too, but I remember it was a lot to adjust to at once, and it made calculus seem magical for a while, rather than logical.
Hearing "difference equation" just gave me flashbacks to my second year electrical engineering courses, where we worked with discrete signals and needed to use difference equations and z transforms
Oh yes. I don't miss control theory...
I'm a maths student. Honestly my path is so much easier thanks to interesting CZcams channels like this one. Thanks for that.
My father (a high school math teacher) introduced me to this topic in the mid '70s. Thank you Mathologer for filling in some of the gaps that have developed over the years, and going further than he could with a kid in junior high.
As a math teacher myself, I must say that this is beautifull and it's not very known even at graduate levels; I've loved this difference stuff for many years now and had very few people with whom share talks about it. Greetings from Argentina!
Bro I made machine to calculate differences for any input sequence..
Hey, I reverse engineered the progression of XP required for each level in final fantasy 5, using this technique (I didn't know about the Gregory-Newton's formula but somehow I figured a formula).. I was 13 at the time or something. I wanted to make a game and for some reason the XP progression was going to be a big part of it. (I ended up making a small dungeon in rpg maker)
It turns out that the XP progression is a polynomial, but the last row is a bit random - instead of being the same number (and thus the next one being all zeroes, and the other all zeroes too, etc), eventually it had a random noise of 0, 1 and -1. I figured out that this meant the coefficients of the polynomial weren't perfectly integer, and rounding messed up the differences.
That's so cool! That's real life math you did!
The answer to your question is "YES" . This is the reason why!! You need to understand we live in a magnetic world, EVERYTHING IN THE UNIVERSE IS RELATIVE TO MAGNETICS!!! The speed of light is directly proportional to the particular magnetic field it travels through!!!! E=MC2 is nothing more then a JOKE!! E=MD, (M'agnetic D'ensity),
EVERYTHING you see and feel is in our magnetic realm all tree's all plant life all human life, We are all a magnetic entity!
I'm just simply amazed that it never occurred to me that solving "next element of sequence" puzzles on IQ tests is actually just applying differential calculus. I guess I just never gave it deeper thought...
Amazing video, great explanation :)
I hope you've gone back and claimed those 20 - 60 points bub 😸
This was one of the most ah-ha-dense videos you've made. Long time fan; this was one of your best. I came away thinking about an entire mathematical space I hadn't thought about much before.
A very AHAmazing video
I was actually tossing up whether or not I should leave out some of the AHAs because the video was really getting quite long :)
You really did keep that self contained. I understand the principles of calculus from school but I never realised how beautiful mathematics was, mostly because I had undiagnosed ADHD so something that required extreme focus and care like mathematics just became frustrating because I would always seem to make silly or seemingly careless mistakes and so I just avoided it. Thanks for your videos!
Glad this works for you ;)
Okay..Half way through the video..They do teach this to us..In highschool..But not in calculus..But in Sequences and Series under the name : Method of difference..Our teacher said.."When u find no other way(like the classic Vn method as they call it here) to find the General term of a sequence..Use this method to find it"
You should ask your teacher whether he or she is actually aware of the calculus connection :)
@@Mathologer I am 90% sure they won't know or even the proof for it. The original poster has an Indian last name [sarkar meaning goverment]. So he is probably preparing for jee in a coaching. The method of teaching in coachings is highly utilitarian because the exam is highly competitive and not a single second is used on something which won't be directly useful in the exams. Only how to solve questions is taught.
Somewhat sad but this is what high competition for resources does. Deriving proofs yourself turns detrimental . I think this harms scientific aptitude of India. Though it does teach working under pressure and being extremely practical while avoiding perfectionism if someone does pass the exam.
@@pravinrao3669 That actually sounds similar to Greece! It must be different in it's ways, for example we all take private tutorials during cenior high school although it's supposed to be optional.
We had though one math teacher in cenior high school who taught "useless" things and made fun of tutors and shared math books. Most tutors and students hated his guts but I love him! "The student says I'm tiiired! I solved 30 exercises! And what got tired? His hand got tired. Not his mind. He solves and solves and his mind shrinks and shrinks... Open a University book, to open your mind!"
Sequence Calculus is just low resolution calculus.
I’d say approximate calculus. Calculated calculus. Computed calculus. No, wait! Digital signal processing! That’s what it is. DSP.
I remember noticing that the 2nd difference of the square numbers is constant and the same for the 3rd difference of the cubes when I was about 12 in school, so this is weirdly nostalgic. I've never learned about any of these details though! Very cool
Not just that the 2nd and 3rd differences etc are constants, but the factorial of the power in question. :)
Back in my early teens I was fascinated by this stuff. I'd spend hours re-discovering all these relations (nobody taught me that in school, but I stumbled on it myself by some chance). Thanks for the fun reminder 😃
"Whatever you want comes next"... the sentence that blew my mind 🤯
Definitely a great life lesson :)
The audio has definitely gotten louder, and I appreciate!
Hi! My name is Alex 15 years old and I'm a big fan of yours. I discovered the exact same formula about 2 years ago while... well let's say I had too much free time. I was really really excited to see that you've made a video about it. And thanks for the proof. I kinda missed that part when I showed it to my classmates by the name " I can guess your polynomial ".
Hi you are my best friend
@@mger2065 dude...
It’s also cool to imagine how you could increase the depth of the mystery sequence’s differences to give it an even longer deceptive start. Now that would *really* confuse people.
Damn, this is one thing I've know about for a while but no-one else seemed to know about, so thank you for showing this to the world.
Soon we will have all of your special knowledge.
It's relation to factorials can you explain
I actually remember that I discovered this accidentally back in high school when I wrote out 1,4,9,16,... and then the differences between each number. I did it just for fun. And I noticed that it behaves exactly like the derivative. I was amazed by that but my interest didn't go further than that.
That's amazing. I found it out when I was trying to learn more about what makes the difference between the different powers and did it for ^2, ^3, ^4, and ^5
Same bro, I did that too
Same here lol. I found that the powers terminated and I can reconstruct the next number in the sequence by just adding the terminated "constant term."
If this doesn't deserve a like I don't know what else does. You just get intuition from this kind guy for free. His channel is amazing.
This is EXACTLY how calculus was introduced at my school (at the time). I’ve always found it very intuitive. Great to see this explained here in such clarity!
Ah yes, I gave a talk to the math club at my school demonstrating the difference calculus applied first to polygonal numbers, then I revisited the triangular case and derived the tetrahedral formula, ending with a pascal triangle surprise. The interplay between the discrete and continuous is, in my opinion, an understated mathematical motif which I felt the need to highlight in my one and only pedegological presentation.
I think mathologer deserves millions subscribers .
I was learning about “discrete calculus” like this in my undergrad and I was blown away when I learned that 2^n is its own difference, much like e^x is its own derivative. It’s like 2 and e are analogs of each other.
I’m still trying to grapple with the mathematical-philosophical implications of discrete calculus being epitomized by “2” and continuous calculus being epitomized by “e”. They aren’t that far apart on the number line, after all.
Nice, isn't it? I'm thinking this has to have something to do with the fact that 2 == floor(e).
I think i figured out why they are so close together. In fact why the difference calculus "e" must be less than e. Using limit definition to differentiate a^x you get a^x lim h->0 (a^h-1)/h. Let a=e for then we get lim h->0 (e^h-1)/h=1. So the secret is the function lim h->x (f(x)^h-1)/h)=1. If you exclude 0 from domain, f(x)=(x+1)^(1/x) which if you take limit to 0 is "e". f(1)=2 as desired. f(x) is monotonically decreasing and limit is 1 so whatever "e" is in difference calculus must be between 2.718.... and 1 so it's not surprising "2" and "e" are close. unfortunately i dont think floor function has anything to do with this.
Also, if you want to look at f(2)=sqrt(3), that means the "e" in difference calculus where you take difference of ith and i+2th term other term is sqrt(3). Which could arise some more interesting math.
The analogy boils down to this:
e = (1 + 1/n)^n for n -> infinity
2 = (1 + 1/n)^n for n = 1
@@rasowa2958 here is a ⁿ to use in place of the ^n
I think this is the most useful Mathologer I've seen yet. TY!
I was so excited when I found this pattern with the exponents when I was younger
Edit: yay you went over it
I did it for the squares, cubes, fourth powers and fifth, but then I messed up in arithmetic while trying to demonstrate it for the sixth. Still very cool to have discovered it, and it was cool to see the parallels in my first calc class
Me too
Another founder here! I was so excited at the time. I actually didn't know people knew about these stuff before this video 😂 Thank you Mathologer!
It was amazing to discover that all that really had sense
I love playing with differences and sums of sequences! (I had taken to calling them "discrete derivatives" but that might be sort of a backwards name.) So cool to see a proper mathematician talk about them!! You went much deeper than I did, but I did use this on 2^x and x^2 to see how it worked similar to a derivative, and on the Fibonacci sequence to note that it is exponential in nature :)
That the number of rows is the same as the degree of the polynomial feels so nice to me
The name I had learned, was the staircase of a sequence.
Sequence is analogous to function
Staircase is analogous to derivative
Series is analogous to integral
great stuff. the 'evil oracle' version of what's next is an interesting logical idea. every time you guess, he says "no, that's not it" and gives you another number in the sequence that doesn't match your guess.
Bringing something completely new to my attention is why I love this channel.
The last time Mathologer posted a video the night before my exam, I got selected even though I didn't prepare much. I'm hoping that the trend continues. :D
Seriously though, the timing couldn't have been any luckier.
Selected in an institute no less prestigious than ISI, one of the best maths institutes in India.
@@spiritbears wtf bro..He's talking about Indian statistical institute..u get into ISI..through RMO nd INMO..And in the rare case u were being sarcastic..M dumb
@@spiritbears yep! I'm appearing but I don't really care about the results cuz as I'm staying in ISI (Indian Statistical Institute).
Continue the sequence "Pass, pass..." :)
Thank you for this new, great Mathologer video! I am really enjoying this long video about that absolutely interesting topic! 😄
Dude that was great! I didn't even know about sequence calculus until now.
Bonus problem: Show that the sequence shown at the start of the video (x_k = 1, 2, 4, 8, 16, 31, ...) is the maximum number of pieces that can be formed by slicing any convex four dimensional polyhedroid using k-1 hyperplanes.
I guess its kinda analog to the 2D-Version right?
Sure my proof is the references for that in OEIS A000127
@@timohuber536 Yes, exactly. If you try it in 1D, 2D, 3D first, you can find 1st, 2nd, and 3rd degree sequences. I think you can build an inductive proof.
For those familiar with VC-theory this sequence also appears as the growth function of the perceptron, since a trivial perceptron is just what people call a "linear separator".
And the neat thing is that you can generalize the discrete approach to several dimensions, and talk about heat equation on grids... But why stop there? Isn't a grid like a special case of a graph?
Like the plane, that is a very particular case of a manifold?
With some reasonable hypothesis, many theorems have a parallel discrete contrepart!
That was the topic of my master's degree, btw. And, just because maths is full of interesting connections, studying this is equivalent to studying random walks on graphs AND everything can be paraphrased into electrical network terms!
Hmm.. supervised machine learning is about finding multidimensional functions from some multi-dimensional data points.
Would you say that that this method could do what machine learning does (assuming perfect data)?
ps. "perfect data" is of course a big assumption, because machine learning approximation averages away bad data, while a "sequence" may not.
pps. How do you define a "sequence" if you have several dimensions? What's first, 2,3 or 3,2?
@@acasualviewer5861 the approach is quite different with respect to ML. Here, the graph is a given, and mostly, the fact that each point has some neighbors at a given distance; in ML, there isn't a priori a natural link between points in the dataset: it's the purpose of ML defining that.
As you already realized, in more "dimensions" sequences aren't enough, so you use different operators to encode the "derivative" concept: I worked with the analogous of the Laplacian.
If you think about it, it's similar to when in multivariable calculus the directional derivative isn't enough, and you study the gradient.
@@Kishibe84 well there's always time series data. But yeah.
I just find that this way of defining functions could be applied in ML when you have very few data samples.
I wonder if it could work with vectors.
I covered finding the equation for a sequence in the way you did for the Fibonacci sequence in my discrete structures class, but the reason it worked wasn't well explained at all. Seeing it in context makes the process so much more sensible and natural! Thank you for this awesome video.
This may be the coolest video I have ever seen on any subject. I love series in the first place but this just pushes it to a new level for me. Thank you so very much for these new insights.
Truly one of the best maths content creators on CZcams, your work is outstanding
Hey! I've seen your videos since last year, and I really enjoy it. I turned 16 couple days ago and I'm really used to studying for olympiads. In fact, I was one of the 4 people from Brazil to go to "Conesul", it's a south America olympiad, and I'm studying really hard to go IMO. You and 3b1b are the only foreign channels I know that make videos about the "real math", and I truly love watching your videos. And my request is, would you make a video solving the problem 6 from the 1988 IMO? It's a very famous problem and I'm sure you know the problem and the solution to it (me too btw), but I would love to see a video of yours solving this problem. Jokes aside, I would watch it every morning lol
The next video has a bit of an IMO angle. If nothing goes wrong I'll put it up either this coming weekend or the next :)
3:40 After you had got to the line of 1's I immediately got to my mind what Babbage was trying to do with his machine. Automate the process of calculating differences. So this is calculus because calculus is the mathematics of differences.
Great enriching video, worth the time.
Nice pace. I like to rewatch excerpts whenever my mind wanders off from the understanding solution.
Thanks for posting the presentation.
so beautifully laid out, the examination and the best explanation I've ever heard.
Of course, it's not taught, but as long as one thinks, not teaching this won't stop a few, very thoughtful students
from discovering this method.
They never tell you how to win. You must learn.
Maybe you can use 1112111178 or 10009006. Or learn this hard math.
@@jmk527 Yep, that's the result of teachers forcing students to obey with grades at the cost of their jobs,
which in turn, costs their futures, whether it be a poor life or a very premature death.
This is why if students learned the same material outside the system, they can dramatically decrease or fully eliminate the worry
of getting something wrong whilst avoiding the cost of their futures.
¡Viva la revolución!
@3:00 powers of 2
The first powers of 2 that appear (1, 2, 4, 8, 16) correspond to sums of complete rows of Pascal's triangle. The 256 corresponds to the sum of the first half of the 9th row of Pascal's triangle. There are also the numbers 64, 16, and 4 in the next three rows.
There seem to be a few more powers of two hidden in the sequences for the lower rows, but I wasn't able to find another power of two for the main mystery sequence.
Calling the bottom row of ones f_0(n), and then the higher rows f_1(n), f_2(n) etc. with the mystery sequence being f_4(n):
Here are the powers of two I understand:
f_k(n) = 2^n for k
Amazingly nice analysis, thanks for sharing! I actually don't know myself whether there are any further powers of 2 in the original sequence :)
Great job!
I did stop way before this, using only an open office table, because: if it is not obvious there, it is miles beyond my possibilities...
It heartening, hilarious, and humbling to see the various people reporting "I discovered this when I was much younger" (or a variation on it). I am fascinated by math and terrible at it. So I was 26, did everything by hand without a calculator (not because I'm hardcore, but because that was all I had), and twice ran into worsening and worsening polynomials to try to deal with, before I finally came up with a one-line formula that spat out the polynomial for a sequence of numbers. In this attempt, I started over twice and I don't know what I did differently the third time (when I say I started over twice, I mean over months of hand calculations). I certainly had no names for the sums of sums, but I did see Pascal's triangle buried in it, was multiplying the falling "n choose k" statements by coefficients derived from differences of differences. I was using horizontal rows, rather than diagonals though.
Watching the video, I can't put together the relationship between what's happening here and what I did exactly; this is obviously much more straightforward and sensible, but both end with elegantly simple statements. I have a really ugly Excel spread sheet (the "function finder") that crunches a list of numbers into a polynomial. The background calculations are just as ugly as my stumbling attempts, but it gives you the correct polynomial.
Just to show the prettiness, if you have a sequence of four numbers [A, B, C, D] that generates a degree 3 polynomial (variable n>3), the not-simplified formula that kicks out is (I think):
[-(n)(n-1)(n-2)(n-3)]/3! * [A/(n) - 3B/(n-1) + 3C(n-2) - D/(n-3)]
Much complex, yet still following. These videos are truly amazing!
Coming from computer development, where the three most common problems are "naming things and off-by-one errors"; this indexing system makes great sense and I think it should be more common, so as to demythify the zero condition
THANK YOU! This is a fantastic presentation you created here... with easy steps to follow the logic. I personally consider Newton the greatest scientist/mathematician of all time ... Tesla second. This presentation shows what level he thought on ... and consider he was in his late teens and early 20s when most of this was formulated.
I am from India. Firstly, I watched your video of Ramanujan's sum and then MASTER CLASS of power sums, I became a fan of yours. Your videos are one of the best animation and entertaining explanations of complicated topics.
Thank you so much for your efforts🙏🙏
I agree. I mostly watch his MASTER CLASS -videos, and have learnt tons of cool, new stuff, from them. By the way, Ramanujan really was a Visionary, with a Capital ”V”. Even most Savants, I’d argue, can’t compare to him; and he lacked most of the formal Mathematics-education, of his time. 😮
P.S. Lots of love and respect to India, from Finland 🇫🇮❤🇮🇳. 😌
I just watched this video for the fourth time. And I’m most likely going to watch it many more times. Not because I have trouble to grasp it, on the contrary.. it’s like a master piece of music. Plus, in every view I get some new inside I didn’t think about before.
Than again, this is the common characteristic of all of your videos. Thank you so very much.
Great video. Thank you very much for revealing something that was never taught to me in my calculus classes!
I only remember some of this vaguely from university. They glossed over most of the calculus of differences. Although when you get into Lebesgue integration and measure theory they just sort of start assuming you know this stuff already.
All videos worldwide in all social media should be as educative as yours.
Thank you 😊
Wonderful! Spectaculary clear and teacherly. Richard Hamming's book "Numerical Methods for Scientists and Engineers" (a staple in engineering education 50 years ago) discusses "Difference Calculus" and "Summation Calculus" and describes "Summation by Parts" (!). For those of us who wrestle with numerical problems, all this material provides powerful tools and the basis for software algorithms that do the heavy lifting in science and engineering.
Another of my treasured tomes.
I managed to stumble upon independently Newton-Gregory using linear algebra and Jordan canonically form while searching for an expression for a sum of N^3, there lies quite a detailed and insightful derivation in Linear Algebra .
My favorite math teacher is back let’s learn a new thing :D
I have been using Newton interpolation to interpolate between numerical differentiation steps (in Fortran). I did some study of difference methods and your video just adds so much more insight. Thanks very much.
4:43 I love how the smile is in the subtitles too lol
It would be fun if WolframAlpha implemented a "what comes next"-function which uses the Gregory Newton Formula along other algorithms or databases (like the OEIS) to predict the likelihood of the next upcoming integer :)
You can use polynomial interpolation to predict the next integer for pretty much any sequence. I tried it in python with the sequence of the start of the video and it worked perfectly.
They do.
@@sergioperez1543 This differences is just polynomial interpolation! If it becomes constant in 2 steps… it’s has a maximal term of n^2 and is thus a quadratic.
Another great video as usual, I will definitely revisit this one after doing more studies. I just love the attitude towards math you have, and that you poke fun at those "intelligence tests". Those tests are so easy if you just know the trick, it's hardly an 'intelligence' test as it is a knowledge test at that point. Maybe very clever people might come up with the solution with no prior knowledge though.
Beautiful! This is exactly what I was trying to show in my comment to your Moessner Miracle video, that summing a sequence is roughly equivalent to integrating the sequence.
Another master class video from mathologer. Your videos make maths so cool.
All this ending 1s really tickle my Collatz conjecture obsession
I really like the design of your shirt :)
Nice video, as always! Really interesting topic; I once thought about the sequence difference being similar to the derivative, but I didn't formalize it. Glad to see it was a thing!
Like that shirt!
Sieht echt cool aus, danke für das tolle Video!
Thanks for another interesting video! One question though: In the 3rd charpter, about 28:30 min in the video, in the difference scheme of the squares, shoudn't the first differences (the second row) be 1,3,5,7 instead of 1,3,7,9?
Well spotted :)
36:19 The x's on the right-hand side of the second row are supposed to be n's.
Thanks for the wonderful video! I think this would be a wonderful approach to teaching calculus in school. Start with differences and sums, and then go over to differentials and integrals. Just like we teach probability theory, even in college: We also usually start with discrete probabilities and then go over to continuous concepts of probability.
thanks Mathologer! Absolutely stunning. You make things so accessible. And showed the beauty of the math so elegantly.
We love these long videos.
What's really ironic is that when I was studying engineering, after 3 years of teaching me how to solve differential equations, they finally admitted that real-world engineering problems rarely have closed-form solutions. So that's when they started teaching numerical methods like Finite Element Analysis and Computational Fluid Dynamics!
This video-which drew a parallel between sequences of discrete numbers and continuous functions-reminded me of that!
Calculus is soft, silky, continuous, and exact. Numerical methods are rough, gritty, discrete, and approximate. I think that's why math departments continue to teach contrived exercises that resolve to beautiful, closed-form solutions. Whereas engineering departments are forced to tackle real-world problems head-on. Engineers need to arrive at reasonable solutions that are practical to implement; they need not be perfect.
The reason why they teach the continuous/analytical stuff is because it is still useful, and one should understand how to derive actual, precise solutions even if one only works with discrete approximations in practice. Hopefully analytic solutions for those problems which currently have no known such solution will be derived/discovered in the future.
As George Polya amusingly but somewhat truthfully said on the topic of PDEs: "In order to solve a differential equation, you look at it until a solution occurs to you."
Perhaps as an engineer or someone in your particular field, analytic solutions are not useful to you - that's fair enough - but they are still useful, and the practice/art of finding them is still useful as a whole to mathematics. Just as solving the heat equation led Fourier to devise Fourier series, so too may further attempts to solve other problems yield other techniques with myriad applications.
@@JivanPal You and I are in far more agreement than you think! 😉 I agree with all the three points you made:
1 - By all means, analytic solutions should continue to be taught to all students of STEM and related subjects. I did _not_ suggest that it should be abolished! 😆
2 - Researchers too should continue to study in detail the properties and behavior of systems.
To cherry-pick another example, the Navier-Stokes Equation (NSE) is widely used in certain fields of engineering. Unfortunately, it doesn't have any known closed-form solutions. So engineers solve NSE problems by either simplifying the equations, or numerically, or both. Further, many general properties of the NSE are unknown as well. We don't know if solutions even exist in many cases, or if they are continuous, stable, well-behaved etc.
Not only will engineers be eternally grateful to the researchers who solve this problem, there's literally a million-dollar prize from CMI to whoever solves it first.
3 - However, practitioners (including engineers) must be given the training that's most relevant and practical to them. And *excessive* focus on analytical solutions is not productive (the key word being "excessive"). They need to finish their studies in a reasonable amount of time and go out into the real world and start solving real-world problems!
For example, consider the typical problems we solve: Heat flow, fluid flow, stresses and deformations, electric and magnetic fields etc.
With only analytical techniques available, we were able to solve them only for the simplest of geometric shapes. It was only after we were taught numerical techniques that we could solve those problems for real-world, irregular shapes like heat sinks, turbine blades, crane hooks etc.
In the past-before digital computers-solving complex systems numerically wasn't an option. So engineers simplified equations as much as possible, then tried to solve them analytically. Accordingly, engineering courses too placed a heavy emphasis on solving PDEs analytically using pen-and-paper.
My dad-who's also a mechanical engineer, but a few decades older than me-belongs to that generation. To them, _"Assume a spherical cow"_ 🐄=⚽ is not a joke; that's literally how much they simplified their problem statements! 😂
But mind you, he too wasn't taught all known solutions to all known PDEs-only a few selected representative techniques. For solving actual problems, he invariably looked them up in his _Handbook of Mathematics, Volume 5 - Partial Differential Equations in 2 or more variables._ If the solution wasn't there, he simplified his equations and repeated the process.
By the time I entered engineering college, computers were already being used for engineering. While I was doing my first-semester coding assignments in the computer center, I saw seniors designing complex shapes using Ansys and other such software. Furthermore, computer algebra systems were also becoming more and more powerful. So you could just enter your equations into one of these tools, and it would either spit out a solution, or hang if it couldn't find one in a reasonable amount of time. 🙄
Today it's impossible to do engineering without computers. So based on my own engineering career, I'd say teaching engineers more software tools and less theory would pay bigger dividends. Again, I didn't say "zero" theory; just a little less than what I was taught. Obviously, teaching the fundamentals is mandatory-that goes without saying. And equally obvious, the proportions are different for different professions.
@@nHans, I totally understand and agree with everything you've said. It is surprising, then, to me, that you feel you were excessively taught analytical methods. At least in the western world, emphasis on numerical methods in relevant fields is the norm, e.g. first year undergraduate physics and mechanical engineering students get right to working with tools like MATLAB and Python to solve problems using methods such as finite difference and successive approximations from the very start.
It'd be interesting to hear more about your experience while studying, and where you did so.
@@JivanPal I completed my mechanical engineering degree in India in the 1990's. 👴
We did have computers back then, but our professors were much slower than the West in adapting to the brave new world. They continued to teach the same curricula they had been teaching unchanged for decades. _"Why fix something that has worked very well in the past?"_
Have you read Richard Feynman's _Surely You're Joking, Mr. Feynman?_ In particular, his critique of the education system in Brazil? India's was exactly like that. In fact, that's what I said when I read that bit: _"Hey, our Indian education system is exactly like Brazil's!"_
However, while my original comment was indeed based on my personal experience, I've seen similar problems in other countries too. I worked for several years in the US. During that time, I interviewed a fair number of job applicants, including recent graduates. I found the same problem (though perhaps to a lesser extent than in India): There are big gaps between what colleges teach and what industry expects from engineers. And college courses are always (perhaps inevitably) a few years behind the state of practice.
Now we are just waiting for Mathologer to compare the Laplace transform to the Z-transform in a simple way :)
That would be an amazing feat
Big fan from Greece here, thanks for the shout-out! :):) By the way, a friend once asked me "you actually use greek letters in everyday language? They are not just for math?" :D:D
Awesome video! I appreciate the effort you put in to these!
5:00 "Fasten your mathematical seatbelts" is one of yours (and mine) favorite phrase. But how does a mathematical seatbelt look like ? Perhaps like a giant string made of your amazing T-shirts ?
A seatbelt twisted into a mobius strip
Why everytime that I'm intersted in a specific branch of mathematics Mathologer does a video explaining exactly what I was searching for?
2^n being parallel to e^x blew my brain, the whole video is just a testament to elegance of mathematics.
Well, 2^n is just the first step on the way to e^x when you use the (1 + 1/x)^x formula to get e.
My first time here, why I didn't hear about this channel before? awesome channel, subbed instantly, love from Jordan.
I'm more and more convinced that Pascal's Triangle is the heart of mathematics.
Yea me too
Yes, it's the triangular hear of mathematics :)
Great video! I have been watching these videos since I was in high school, and now I'm at the end of my degree in mathematics! One of my favorite topics is set theory, notably ZFC, which you are most likely familiar. Is there any chance for it to be covered in the future? It's a very mathy technical topic, but once we get the gist of it, it's definitely perplexing. Cheers!
So many great things to talk about so little time. This topic is sort of on my to do list, just not sure whether I'll ever get around to covering it :)
THIS is the Calculus video I was looking for. THANK YOU SO MUCH.
I appreciate the coverage of difference calculus very much. Excellent treatment of the topic. Helped me. Thanks. Subbed.