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..
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!
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
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.
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!
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!
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) th-cam.com/video/VLbePGBOVeg/w-d-xo.html
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. 🙏🏼
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.
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...
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!
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.
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 )
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.
@@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!
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.
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.
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!
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.
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.
@@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?
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.
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!
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.
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 :)
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.
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!
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 🇫🇮❤🇮🇳. 😌
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.
That was fascinating. By the way, bottom row is all ones 1, next row is natural numbers n, row after that is 1 more than triangular numbers or central polygonal numbers n(n+1)/2+1 OEIS A000124, row after that are cake numbers (n^3+5n+6)/6 OEIS A000125, finally last is 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 (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24 OEIS A000127. Great video as it's far more satisfying to derive them than cheat with OEIS though it contains a wealth of interesting facts, recursive formula, etc
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 ".
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
@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
A teacher is not a person who understand a subject well, but one who can explain it. Amazingly done. I rewatch the video over 5 times. Every time I rewatch I learn new thing. Thank you,
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
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.
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!
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.
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
Gotta love the consistency, as well, that the difference is marked with a Greek Capital Δ, while the antidifference / sum is marked with a Greek Capital Σ. 😅😌👍🏻
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.
sir i am in love with your way of thinking maths, i have searched for someone like you so much in my school and got hit in the face or abused, i am so happy that people after me can relish on the beauty and pleasure of mathematics less challengingly and more powerfully through your contributions through this channel on youtube, I AM INDEBTED TO YOU IN MORE WAYS THAN I CAN EXPRESS. THANK YOU SO MUCH!
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.
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
13:55 polynomial sequence 15:30 Lagrangian interpolation Builds poly formula in 1 easy step (me: nothing's easy about it) 15:38 Gregory-Newton formula (me: n choose k method hehehe) A permutation -see combinatorics 20:35 what's next formula solution concept is about 1. inferring (distributions) of a naturally occurring formula 2.guess the natural general rule of the outcomes 3. Justify why Guess is correct (& not other random number) 20:58 have to prove formula "Correctly Captures" the (right) mathematical Context I. E. Lines connecting Paris of dots Cutting cicrcle into regions; Number regions depends on Location(dots) N dots = n regions 24:08 Del(exp(x))= pow(2,n)= sum(n choose k)
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.
@@pranavtiwari_yt 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
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
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.
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)]
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 😃
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"
@@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!"
This video has allowed me to finally pin down the annoyance I feel when I see one of those 'complete the sequence' puzzles: "There are infinitely many functions containing this sequence, and you expect me to read your mind and come up with the one that YOU thought of?"
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.
2:57 “Hardcore” Challenge I wrote a quick Python program to calculate the terms of this sequence for me. After not finding any additional powers of 2 in the first few dozen terms, I set it up to automatically check for powers of 2 as well (and only print the powers of 2, so I don’t have to look through all the terms) Unfortunately, it seems that after calculating out to one billion terms, there are still no additional powers of 2. The only terms it prints are a_0 through a_4 (which are 1 to 16), and a_9 = 256 (There is no integer overflow in Python, so it’s the perfect language for quick mathematical investigations like this, even out to a billion terms)
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.
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.
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.
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.
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
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!
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
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 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.
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.
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.
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.
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.
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 .
I learned some of this during my senior design project while taking engineering classes, where i realized that differences were a simple stand in for derivatives for real world problems.
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.
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
I first learned about the calculus of finite differences from reading Martin Gardner. I had no idea how rich the subject was. The Newton-Gregory formula is a real gem.
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.
@@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".
When I was in school, I followed the "Kumon method" to improve in mathematics (it's basically a method invented by a Japanese guy that consists in doing A LOT of excercises that increase slowly in difficulty. You tipically start doing 1st grade sums and can end up solving some problems involving integrals). This method actually taught how to do this discrete calculus somewhere along the line, and it was rather easy to understand because it showed how to derive the original sequence from the difference sequence: when you sum the terms you end up with a telescopic sum, so it always works. Needless to say, this technique was very useful and I ended up using it for every math problem involving sequences that I found, it worked like 90% of the time (The other 10% the sequences involved some sort of geometric sequence, so you could just do the same with quotients).
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 ?
50 years ago when first learning numerical methods, I discovered that you could represent the derivatives of a sequence of numbers using finite differences. If you put the differences into a square matrix (ignoring the one leading or trailing value) you had a dandy way of manufacturing the derivatives using simple linear algebra. It occurred to me that the inverse might give the integral. It did! Modifying the matrix for a periodic sequence, the inverse failed, being of rank 1 less than the number of terms in the sequence. It dawned on me that this was a restatement of the fact that the integral represented by the inverse of the matrix did not have the "c" we sometimes forget. Very cool video - enjoyed every bit of it.
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.
@@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.
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.
@@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!
Great video. I had never thought of it that way even though I have probably read it and forgot its beauty. I also advocate student to do these "discrete derivation" steps and see if they get to a row where all values are similar and don't understand either why it's so forgotten. It's such a great problem solving methodology. Then why it is a indication that the points might be a polynomial of k:th power, where k is how many times they have "derived". I usually take the sum of squares as first example, close to what you did. I have on the other hand told them to set up the desired polynomial as ax³+bx²+cx+d and then replace x with the order the number have in the sequence. I usually include 0 so to be nice. This will make them get the four equation d=0 a+b+c+d=1 8a+4b+2c+d=5 and 27a+9b+3c+d=14 which they then solve with Gauss Elimination making a=1/3 b=1/2, c=1/6 and d=0. So I sort of make them do a vandermonde matrix without them even noticing. They will have to make the indexing correct though our their polynomial might get transformed on the x-axis a bit, but would still be able to predict upcoming values. Probably easier with the method you suggest here when it's an even bigger power though as the Gauss elimination or matrix inverse processes are prone to introduce accidental mistakes by students and myself :). Thank you so much for the insight! Maybe even better even for lower powered polynomials :) It all depend on the circumstances! But this is more beautiful I agree. Another nice example of this potential is make this exercise: Find the second-degree polynomial that intersects these points (2,1) (3,6) and (5,28) making them realize we can set up such a matrix even if we are "missing" a value if we know n+1 points and that we are seeking a polynomial of the n:th power. Your channel is really great as you put that little extra concise effort in the sentences and phrasing, making it unlikely for viewers to take it out of context, which seems to be a big problem on TH-cam nowadays.
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?
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.
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!
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) th-cam.com/video/VLbePGBOVeg/w-d-xo.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'm a maths student. Honestly my path is so much easier thanks to interesting TH-cam channels like this one. Thanks for that.
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 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...
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.
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.
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 )
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!
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.
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.
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..
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.
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?
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.
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!
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 :)
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 😸
I think mathologer deserves millions subscribers .
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
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 ;)
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 🇫🇮❤🇮🇳. 😌
"Whatever you want comes next"... the sentence that blew my mind 🤯
Definitely a great life lesson :)
Truly one of the best maths content creators on TH-cam, your work is outstanding
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 :)
That was fascinating. By the way, bottom row is all ones 1, next row is natural numbers n, row after that is 1 more than triangular numbers or central polygonal numbers n(n+1)/2+1 OEIS A000124, row after that are cake numbers (n^3+5n+6)/6 OEIS A000125, finally last is 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 (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24 OEIS A000127. Great video as it's far more satisfying to derive them than cheat with OEIS though it contains a wealth of interesting facts, recursive formula, etc
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
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...
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 think this is the most useful Mathologer I've seen yet. TY!
@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...
A teacher is not a person who understand a subject well, but one who can explain it.
Amazingly done. I rewatch the video over 5 times. Every time I rewatch I learn new thing.
Thank you,
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. :)
Bringing something completely new to my attention is why I love this channel.
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!
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.
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
Gotta love the consistency, as well, that the difference is marked with a Greek Capital Δ, while the antidifference / sum is marked with a Greek Capital Σ. 😅😌👍🏻
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.
sir i am in love with your way of thinking maths, i have searched for someone like you so much in my school and got hit in the face or abused, i am so happy that people after me can relish on the beauty and pleasure of mathematics less challengingly and more powerfully through your contributions through this channel on youtube, I AM INDEBTED TO YOU IN MORE WAYS THAN I CAN EXPRESS. THANK YOU SO MUCH!
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 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
The audio has definitely gotten louder, and I appreciate!
13:55 polynomial sequence
15:30 Lagrangian interpolation
Builds poly formula in 1 easy step (me: nothing's easy about it)
15:38 Gregory-Newton formula
(me: n choose k method hehehe)
A permutation -see combinatorics
20:35 what's next formula solution concept is about
1. inferring (distributions) of a naturally occurring formula
2.guess the natural general rule of the outcomes
3. Justify why Guess is correct (& not other random number)
20:58 have to prove formula "Correctly Captures" the (right) mathematical Context
I. E. Lines connecting Paris of dots
Cutting cicrcle into regions;
Number regions depends on Location(dots)
N dots = n regions
24:08 Del(exp(x))= pow(2,n)= sum(n choose k)
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.
@@pranavtiwari_yt 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
@@pranavtiwari_yt 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..." :)
All videos worldwide in all social media should be as educative as yours.
Thank you 😊
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 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!
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)]
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 😃
I have no idea what is going on but I can feel this guy's passion in teaching so I had to subscribe
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!"
This video has allowed me to finally pin down the annoyance I feel when I see one of those 'complete the sequence' puzzles:
"There are infinitely many functions containing this sequence, and you expect me to read your mind and come up with the one that YOU thought of?"
Thank you for this new, great Mathologer video! I am really enjoying this long video about that absolutely interesting topic! 😄
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.
My favorite math teacher is back let’s learn a new thing :D
2:57 “Hardcore” Challenge
I wrote a quick Python program to calculate the terms of this sequence for me. After not finding any additional powers of 2 in the first few dozen terms, I set it up to automatically check for powers of 2 as well (and only print the powers of 2, so I don’t have to look through all the terms)
Unfortunately, it seems that after calculating out to one billion terms, there are still no additional powers of 2. The only terms it prints are a_0 through a_4 (which are 1 to 16), and a_9 = 256
(There is no integer overflow in Python, so it’s the perfect language for quick mathematical investigations like this, even out to a billion terms)
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.
Much complex, yet still following. These videos are truly amazing!
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.
Because it's valuable and worthwhile.
Don't expect anything of value to be even "talked" about at a public school.
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.
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.
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
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!
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."
I learned calculus this way more than ten years ago, it was extremely helpful.
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 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.
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.
I'm learning differential equations right now and I'm very happy to say that at 34:50 you said exactly what I expected to hear.
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.
Sequences are always fascinating .Amazed you can convert it into a formula.
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
Another master class video from mathologer. Your videos make maths so cool.
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 .
I learned some of this during my senior design project while taking engineering classes, where i realized that differences were a simple stand in for derivatives for real world problems.
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.
THIS is the Calculus video I was looking for. THANK YOU SO MUCH.
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 :)
I first learned about the calculus of finite differences from reading Martin Gardner. I had no idea how rich the subject was. The Newton-Gregory formula is a real gem.
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".
When I was in school, I followed the "Kumon method" to improve in mathematics (it's basically a method invented by a Japanese guy that consists in doing A LOT of excercises that increase slowly in difficulty. You tipically start doing 1st grade sums and can end up solving some problems involving integrals). This method actually taught how to do this discrete calculus somewhere along the line, and it was rather easy to understand because it showed how to derive the original sequence from the difference sequence: when you sum the terms you end up with a telescopic sum, so it always works. Needless to say, this technique was very useful and I ended up using it for every math problem involving sequences that I found, it worked like 90% of the time (The other 10% the sequences involved some sort of geometric sequence, so you could just do the same with quotients).
All this ending 1s really tickle my Collatz conjecture obsession
At 15 minutes, my brain hurts.
Mathologer, thank you for your efforts. May you and yours stay well and prosper.
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
50 years ago when first learning numerical methods, I discovered that you could represent the derivatives of a sequence of numbers using finite differences. If you put the differences into a square matrix (ignoring the one leading or trailing value) you had a dandy
way of manufacturing the derivatives using simple linear algebra. It occurred to me that the inverse might give the integral. It did! Modifying the matrix for a periodic sequence, the inverse failed, being of rank 1 less than the number of terms in the sequence. It dawned on me that this was a restatement of the fact that the integral represented by the inverse of the matrix did not have the "c" we sometimes forget.
Very cool video - enjoyed every bit of it.
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.
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.
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!
Great video. I had never thought of it that way even though I have probably read it and forgot its beauty.
I also advocate student to do these "discrete derivation" steps and see if they get to a row where all values are similar and don't understand either why it's so forgotten. It's such a great problem solving methodology. Then why it is a indication that the points might be a polynomial of k:th power, where k is how many times they have "derived". I usually take the sum of squares as first example, close to what you did. I have on the other hand told them to set up the desired polynomial as ax³+bx²+cx+d and then replace x with the order the number have in the sequence. I usually include 0 so to be nice. This will make them get the four equation d=0 a+b+c+d=1 8a+4b+2c+d=5 and 27a+9b+3c+d=14 which they then solve with Gauss Elimination making a=1/3 b=1/2, c=1/6 and d=0. So I sort of make them do a vandermonde matrix without them even noticing. They will have to make the indexing correct though our their polynomial might get transformed on the x-axis a bit, but would still be able to predict upcoming values.
Probably easier with the method you suggest here when it's an even bigger power though as the Gauss elimination or matrix inverse processes are prone to introduce accidental mistakes by students and myself :). Thank you so much for the insight! Maybe even better even for lower powered polynomials :) It all depend on the circumstances! But this is more beautiful I agree.
Another nice example of this potential is make this exercise: Find the second-degree polynomial that intersects these points (2,1) (3,6) and (5,28) making them realize we can set up such a matrix even if we are "missing" a value if we know n+1 points and that we are seeking a polynomial of the n:th power.
Your channel is really great as you put that little extra concise effort in the sentences and phrasing, making it unlikely for viewers to take it out of context, which seems to be a big problem on TH-cam nowadays.
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 :)
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.
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.
Exactly 🎯!
it's amazing, how people can make something simple, sound complex