Why don't they teach Newton's calculus of 'What comes next?'

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

One volume of Schaum's outlines is dedicated to "The calculus of finite differences and difference equations" (by Murray R. Spiegel) Examples galore!

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

[Mathologer]

Back with another crazy long one. Hope you like it :)

[annaclarafenyo]

Every part of calculus is mirrored in sequence calculus EXCEPT the chain rule. This is what makes infinitesimal calculus extraordinarily more powerful.

[macronencer]

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

[tavishu]

Sequence Calculus is just low resolution calculus.

[vic]

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

[nidalapisme]

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. 🙏🏼

[pavolkollar]

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 )

[bamdadtorabi]

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!

[jimmy]

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.

[UB]

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.

[exponentmantissa]

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!

[ericvilas]

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!

[ianrobinson]

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.

[johnchessant]

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!

[alexpotts]

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.

[mostly_mental]

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.

[toniokettner]

that's why i always hate these what come next questions in "IQ tests"

[erettrrrrr]

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.

[DebayanSarkar]

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"

[Kishibe]

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!