Mathologer - 2021-10-02
Another long one. Obviously not for the faint of heart :) Anyway, this one is about the beautiful discrete counterpart of calculus, the calculus of sequences or the calculus of differences. Pretty much like in Alice's Wonderland things are strangely familiar and yet very different in this alternate reality calculus. Featuring the Newton-Gregory interpolation formula, a powerful what comes next oracle, and some very off-the-beaten track spottings of some all-time favourites such as the Fibonacci sequence, Pascal's triangle and Maclaurin series. 00:00 Intro 05:16 Derivative = difference 08:37 What's the difference 16:03 The Master formula 18:19 What's next is silly 22:05 Gregory Newton works for everything 28:15 Integral = Sum 32:52 Differential equation = Difference equation 36:06 Summary and real world application 39:22 Proof Here is a very nice write-up by David Gleich with a particular focus on the use of falling powers. https://tinyurl.com/ymcyrapz This is a nice lesson from a Coursera course on this topic https://www.coursera.org/lecture/discrete-calculus/differences-k4jBq One volume of Schaum's outlines is dedicated to "The calculus of finite differences and difference equations" (by Murray R. Spiegel) Examples galore! This is a really nice very old book Calculus Of Finite Differences by George Boole (published in 1860!) https://tinyurl.com/3bdjr932 Thanks to Ian Robertson for recommending this one. There is a wiki page about our mystery sequence: https://tinyurl.com/uwc89yub It's got a proof for why the mystery sequence counts the maximal numbers of regions cut by those cutting lines. If you have access to the book "The book of Numbers" by John Conway and Richard Guy, it's got the best proof I am aware of. Here is a sketch of how you solve the Fibonacci difference equation to find Binet's formula https://imgur.com/a/Btu5ZVk Here are a couple more beautiful gems that I did not get around to mentioning: 1. When we evaluate the G-N formula for 2^n what we are really doing is adding the entries in the nth row of Pascal's triangle (which starts with a 0th row :) And, of course, adding these entries really gives 2^n. 2. Evaluating the G-F formula for 2^n at n= -1 gives 1-1+1-1+... which diverges but whose Cesaro sum is 2^(-1)=1/2!! Something similar happens for n=-2. 3. In the proof at the end we also show that the difference of n choose m is n choose m-1. This implies immediately that the difference of the mth falling power is m times the difference of the m-1st falling power. Today's music is by "I promise" by Ian Post. Enjoy! Burkard P.S.: Some typos and bloopers https://youtu.be/4AuV93LOPcE?t=719 (396 should be 369) https://youtu.be/4AuV93LOPcE?t=1709 (where did the 5 go?) https://youtu.be/4AuV93LOPcE?t=2448 (a new kind of math(s) :)
I think I spotted a typo at 28:39: the second row says 1, 3, 7, 9. I believe this was supposed to be 1, 3, 5, 7. This lecture was fascinating, thanks!
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.
32:38 Of course, you can just add to the current area of f(n) half of the difference between the next area f(n+1) and the current one f(n), or average the areas of f(n) and f(n+1); and get the exact area, in between; since the missing areas are always right-angled triangles 😎. (f(n) + f(n+1)) / 2
Newton's work is incredibly hard to understand from a modern perspective. Kepler, Copernicus and Galileo wrote rather easy to understand science texts, but Newton requires constant translation between modern terms and his own way of expressing the same facts.
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!
The exercises in that book are tough. I remember being able to prove or solve very few.
"Discrete calculus" is another way to call it.
It's coincidental that I watched this video today (it has been on my watch later list for a while), since I just got thrown in the deep end of finite calculus in the context of control engineering. Just read through an intro of finite control systems yesterday, where you convert from the continuous time S domain (Laplace transforms) to the discrete time Z domain (Z transforms), and instead of solving differential equations you convert it and solve difference equations.
By setting the correct coefficients (matching system poles, control engineer stuff) you can match the continuous differential equation exactly with a difference equation. With simple linear systems, you can have your complete feedback control algorithm condensed into a single difference equation, much easier for simple digital controllers to calculate quickly, which I thought was really neat. Also, the continuous Laplace transforms analysis become more applicable as the discrete time steps get smaller, which checks out with standard vs finite step calculus.
This just seemed like black magic to me, until this video helped close the gap. Still don't understand the mathematics entirely but this has helped a lot.
the discrete logarithm is just the harmonic sequence
and I'd say also check out John Gabriel's New Calculus channel
Even the great Euler used this Mercator/Gregory/Newton generalized binomial series to obtain infinite series expansions for the exponential and logarithmic functions, leading to Euler's constant and Basel problem solution, including the sums of i^2, i^3 etc. I am your secret student.
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.
Ditto about learning it first, have been trying to remember it's name ever since.
Here in the UK I was taught this before moving on to infitesimal calculus too. But yeah, the link between the kinds of calculus wasn't covered later.
@Sam Webster Yeah, limited at quadratic or cubics IIRC
Well that would have been awesome. Instead, I tended to get teachers and professors looking to show me shortcuts without actual foundation or understanding. This would have actually been useful.
The "downward holding force", imagined as "gravity/graflirty" is all delusion because:
-clouds will never drop/fall much less remain down
-gas: smoke, vapor, and etc. rise all the way up and blend in atmosphere
-balloons rise up even after being thrown and/or held down (no matter what effort)
read great, unbroken, mightful - Antarctic Treaty - few read about:
- No commercial/civilian airplanes are allowed to fly over the circular Ice-Wall surrounding our Flat-Earth (wrongly called: "South Pole" and "Globe")
- No civilian entity allowed to place or set-up business location destination there no matter their intended purpose of locating/moving into there
- No member/nation is allowed to claim ownership of the region, take over it, and build-up and colonize the circular Ice-Wall to make their official
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
@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.
@macronencer My thoughts, exactly. I literally thought: ”Truncation!”, when I first saw that result, in this video. 😅
Wow… this whole finding differences of a set of points of a polynomial is something I figured out in 11th grade. I made this program on my calculator that you input a table of points and it will calculate the area under the curve, and I found as you say, for ‘well behaved nice functions’ it was super accurate. Kinda proud I was able to figure this all out with only knowing very basically calculus at that point in my life.
Cap 💀
I don't remember what I figured out in hs, but in middle school I [re]discovered the geometric distance formula, while in gym class, and used it to program collision detection for a game I was working on. In hs, a fellow calc student, Chris, programmed a game he called penis man on his calculator, while Charles created a 3d maze game lol.
If you can solve independently problems that have already been solved, you can solve problems that haven't yet been solved by anyone.
The answer to your question is "YES" . This is the reason why!! You need to understand we live in a magnetic world, EVERYTHING IN THE UNIVERSE IS RELATIVE TO MAGNETICS!!! The speed of light is directly proportional to the particular magnetic field it travels through!!!! E=MC2 is nothing more then a JOKE!! E=MD, (M'agnetic D'ensity),
EVERYTHING you see and feel is in our magnetic realm all tree's all plant life all human life, We are all a magnetic entity!
I think just the fact that you were interested enough to do it is something to be proud of. Most folks go through life fairly blind to the elegance and beauty of science and math. So - good for you, man. I also did my first ever programming freshman year of college using an HP-41CV scientific calculator. Writing useful tools for the various classes I was in became a standard routine, and I'm convinced it helped me understand the material more completely. I had one that was sort of an "automated Smith Chart" (Smith Chart is a tool used when working with microwave and other radio communication type things). I could have gotten the same answers in a shorter way, just calculating some equations, but I wanted to be able to use it when I was actually dealing with a paper Smith Chart, so it sort of explicitly modeled the steps yo take with a real chart.
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 :)
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.
I’ve watched enough math on YouTube to immediately recognize that first sequence as the number of regions a circle is cut into when n+1 points on the circumference are connected. Without spoiling too much, I would highly recommend an old ThreeBlueOneBrown video titled “Circle Division Solution” to learn more about this problem and it’s brilliant solution.
Thanks. This then gets into lie groups etc? Basically modeling a hyperconnected volume. Super relevant to cosmology
Lol me too
A, the old cakemorphic integers :)
I just wonder how many thousand of years it took a genius or two to note that formulaic progression, and here we have at least two 21st century amateur folks get at it by a few glances? Damn, we’re smart.
I admit unabashed, in orders of magnitude, I’m 0 as a mathematician. A 0 standing on the shoulders of clams.
Most of us manage with knowing any of this. What we have to guard against is treating people who do this as a sort of high priest class, i.e. people with a 'secret' knowledge.
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 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.
Spherical geometry is sometimes called a type of "non-Euclidean" geometry, but actually Euclid wrote a good deal about it.
Are u a qualified actuary?
@Andro Planet I was. I’ve been retired for five years.
@Ian Robinson Certum ex incertis! 😁
👍
This makes me seven more sad that I went into computer science rather than actuarial studies.
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
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!
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. 😮
@Nath S 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.
Oh so it's an equation form of a snitch, so it got stitched
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
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 😸
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 :)
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!
A teacher showed me this back in high school, and I thought it was really pretty. I had wanted to include it in my combinatorics series, but I had to cut it for time. Great to see it covered in Mathologer fashion.
I feel like this sort of "discrete calculus" was something I was vaguely aware might exist, but I've never seen it formalised like this before.
Same, I've been aware of it for quite a while, and even use it constantly to make certain estimations, but never really attempted to study it in a formal way. Well, I "use" it in the sense that just taking the rules from continuous calculus gives good enough approximations for discrete sequences for my purposes.
For example, it often comes in handy when estimating asymptotic time complexity of algorithms. Come across a sum of squares? It's O(n^3), because the integral of x^2 is x^3/3. A sum of 2^n is just O(2^n). And so on... It makes things so easy, and you don't even have to worry about the constants.
I used to doodle when I was little and make pyramids with numbers such as these. It has been like a recurring theme in my doodles. One time I tried to calculate the number of all possible unique trichords in a 12-note equally tempered octave (music theory), and this kind of thing popped up again out of nowhere. It keeps following me
What's better than an "AHA moment"? Over 40 minutes of endless "AHA moment"s, of course!
Loved the video, make more, Burkard! (And preferably faster :D )
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 :)
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 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.
Back with another crazy long one. Hope you like it :)
It's like you read my mind. I mess around with power sequences, and adjacent term differences of them, off and on, and here you are with an awesome video doing that for a general sequence.
Before I get going with the video, if I remember correctly the stuff you did at 4:00 is the nth Forward Difference with Step 1 (n = 1..4, n=4 has all differences at 1).
Hey Mathologer i know you like math problems i got a link to a homer episode where they prove god doesnt exist with math could you check that it is true and why it is
Heres the link https://youtu.be/mGBxUNaQI1I
Most excellent!! Thank you! This lesson provoked an unrelated question I have:
if { [ 0 / (any number) = 0 ] & [ (any number) / 0 = undefined ] & [ (any number) / itself = 1 ] }
then which of the preceding rules is followed to determine the answer to 0 / 0?
I have learned that the answer is undefined, but why does that rule take precedence?
I had to study finite differences when doing actuarial studies. Quite interesting.
what if the difference is constant like in an arithmetic sequence
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.
@rasowa here is a ⁿ to use in place of the ^n
@rasowa →∞ are interesting symbols too
@Akira Kato √
@rasowa More clearly,
for a specific value of h>0 that defines an operator Δₕ such that
Δₕ(f(x))=(f(x+h)-f(x))/h
then we have the identity
Δₕ(aˣ)=aˣ when a=(h+1)^(1/h)
As h goes to 0, we'll have e as the base of the exponential.
2 is an odd prime number because it's even.
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 :)
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
@Timo Huber 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".
Great video. Thank you very much for revealing something that was never taught to me in my calculus classes!
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. :)
Coming from computer development, where the three most common problems are "naming things and off-by-one errors"; this indexing system makes great sense and I think it should be more common, so as to demythify the zero condition
THANK YOU! This is a fantastic presentation you created here... with easy steps to follow the logic. I personally consider Newton the greatest scientist/mathematician of all time ... Tesla second. This presentation shows what level he thought on ... and consider he was in his late teens and early 20s when most of this was formulated.
I 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
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?
@acasualviewer 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.
This feels amazingly profound, A systematic way of looking of analyzing any unknown relationship between elements in a sequence. Something i'd always wanted to know more about but not really able to put into words. Absolutely a gem of a video.
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.
@Rushunnh Fernandes sure
@Rushunnh Fernandes Hello, I am from the Nobel Prize institution. We are ready to ship your Prize. All we need is your address and a temporary deposit of $500 into our bank account. It is a standart security procedure to prevent fraud. We will send it back next day, promise.
@ElDoRado1239 a promise will not do, you should've said pinky promise. Also, Sure I'll deposit the money... But be sure to give me your bank account details.
Will i get a bigger award if I pay 1000 $, the awards are really small.
@PhilBagels Fibonacci sequence is just the Lucas sequence wearing a thin disguise.
That's the reason behind this dislike (that I share too, despite not fitting for the "Ma" + "t" rule). All the f_n = f_(n-1)+f_(n-2) sequences are really the same deal, but people elevate Fibonacci one above others for really arbitrary reasons.
@DukeGarland Like it or not, you can't get away from the Fibonacci sequence, no matter what disguise you put on it. Pick any two starting numbers, a and b. Then you have:
1a+0b
0a+1b
1a+1b
1a+2b
2a+3b
3a+5b
5a+8b
8a+13b
etc. See the Fibonacci sequence in there? It will always be there, no matter what a and b you pick.
Calculus is indeed related to sequences in such a way that the growing sum of the series can be calculated using calculus. The bigger the number the more accurate is the result. This is because of the notion of how we perceive the "unit 1" which we think is "big" that we can manipulate manually at our scale. But the question is how big is "one" really is when there is no unit associated with it? In the same way, we think of zero as having no value but how about if we spread it out with regular intervals over a segment as we always do? Then we can count it and adds up to a certain quantity. Is zero a number or just a representation of "zero-dimensionality" or both? It is the only that when added together will equal itself and no other number. This is because it lacks the 2nd and higher dimensions freedom as we call it. So the smallest interval is 1/∞ (1/infinity) which is not zero. This is the boundary before we dived into zero-dimensionality. But as we can see even the expression "1/∞" still can have a "midpoint" because we associate every point with a number and points are zero-dimensional. Of course, we can fit an infinite amount of zero into a very tiny segment no matter how short that segment is (1/∞). Another thing is that we can also make "infinitesimal" equals 1 as the basic unit of division right? By this, we can be certain that calculus is the one we are talking about when we speak of summing up sequences. Another thing is that "zero" is the basic unit composing "zero-dimensionality" or nothingness I supposed? Then can we accept that "1/∞" is the basic unit composing uni-dimensional line? Or we can just add anything into nothingness no matter how big it is, and does this means that zero-dimensionality is both small (when we shrink an object down to a point) and as big as emptiness when we remove everything including space?
@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 <= n (sums of rows)
f_k(2*k+1) = 2^(2k) (sum of half-row)
Additional powers of two:
f_1(n) =n+1 has trivially all of the powers of two.
f_2(n) = (n^2 + n)/2 + 1 has f_2(90) = 4096 = 2^12
f_3(n) = (n^3 + 5n)/6 + 1 has f_3(23) = 2048 = 2^11
f_4(n) = (n^4 - 2n^3 +11n^2 +14n)/24 + 1 ... after f_4(9) = 256 I wasn't able to find any more powers of two. Maybe I did something wrong in the program I wrote, but I believe I checked for powers up to 2^1000000. (Thank you Gnu MP library.)
Are f_2(90) = 4096 and f_3(23) = 2048 just due to "chance"? Are there any powers of two past 256 for f_4(n)?
Update: There don't seem to be any more powers of two as far as my program has searched. No (n, k) that solve f_2(n) = 2^k, f_3(n) = 2^k, or f_4(n) = 2^k for k less than 2 million. It looks like there are just the 'explainable' ones and then f_2(90) and f_3(23).
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...
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 never knew Gregory-Newton Formula exist. I always wonder how can I get a general formula given the outputs of a function. Can you give us book recommendations for this topic?
Have a look at the description of this video :)
Ah yes, I gave a talk to the math club at my school demonstrating the difference calculus applied first to polygonal numbers, then I revisited the triangular case and derived the tetrahedral formula, ending with a pascal triangle surprise. The interplay between the discrete and continuous is, in my opinion, an understated mathematical motif which I felt the need to highlight in my one and only pedegological presentation.
I 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.
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.
Made me think of Babbage and his difference engine to compute tables of polynomials that approximated functions such as log or trigonometric functions. Babbage had found that tables computed by hand contained mistakes, which caused accidents at sea. His objective was to create tables that were reliable. (The machine was coupled to a printing unit to avoid transcription errors.)
Wow, its just incredible how you always come up with some wonderful new regions or facts in math I never heard before that are always useful or at least very interesting to watch! Also you explaing everything very understandable with the animations, love these aswell. Keep up the great work, you really help a lot :)
I did this on my own in high school when playing with math, but I didn't end my work at the Gregory-Newton formula. Instead, I had it written down as what i called the "sigma substitution" and devised the same scheme to find the successive sums of each row based on pascal's triangle. It's nice to know there's a better way to write it now.
Also, on this same line of thought, I derived the Euler-MacLaurin formula not too long after discovering all the stuff you mentioned in this video by looking for other summations. I'm hoping you'll go over that someday so I can see if I missed anything important there.
I covered the Euler-Maclaurin formula a while back https://youtu.be/fw1kRz83Fj0
ANNA CLARA Fenyo - 2021-10-02
Every part of calculus is mirrored in sequence calculus EXCEPT the chain rule. This is what makes infinitesimal calculus extraordinarily more powerful.
J M - 2022-07-22
Is there an analog of differential geometry?
ANNA CLARA Fenyo - 2022-07-22
@J M Differential Geometry relies on coordinate transformations in it's very essence, in the definition of manifolds, so it would be hard to come up with a discrete analog along these lines. A better way is to use noncommutative geometry as a stepping stone to discrete matrix models of space, this is how space is reconstructed in the BFSS Matrix version of String Theoretic M theory.
Семен Филатов - 2022-08-27
@Pokémon Journeys Fan Mathologer talked about infinite sums. For a sum to converge it should be all zeros at some point.
Similarly, a sequence has a limit if it becomes constant at some point.
Kathir Soft Arts - 2022-09-15
Two way infinity and applications @ kathir Soft arts channel
Comuniune cu Osho - Câmpul Budic - 2022-11-15
are you sure? Check out John Gabriel's New Calculus channel and be amazed