Mathologer - 2019-10-26
The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :) Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only real specialists dare to touch. I worked really hard on this one. Fingers crossed that after all this work the video now works for you :) Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite “proofs without words”, the mysterious Bernoulli numbers (the numbers to "rule them all" as far as power sums go), the (hopefully) most accessible introduction to the Euler-Maclaurin summation formula ever, and much more. Here is a link to a couple of slides that show how to justify having all summations in sight run from 1 to n. This is the challenge that follows the discussion at https://youtu.be/fw1kRz83Fj0?t=2440 . Had this in the video originally and then decided to make this into a challenge: http://www.qedcat.com/misc/change_limits.pdf As usual thank you very much to my friend Marty Ross for nitpicking this one to death (especially for not letting off until I finally inserted that "morph" shortcut in chapter 7 :) Finally, check out the article "Gauss’s Day of Reckoning" by Brian Hayes which tells the story behind the famous story of Gauss adding 1+2+3+...+100 as a kid: https://tinyurl.com/y49buyak Enjoy :) Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer
I have no idea who dislikes videos like this. The amount of effort put into it is just tremendous.
How do you know how many dislikes there are even you wizard
It is a mathematical function for the heuristic of an AI. It's just a math function. It counts towards what you want to see (or not see).
It's not if you think it's a bad video. People who know AI learning know math ... maybe not as deep as this video; but evidently, they know something you don't ... how AI works.
Maybe they hate math
It's 0 dislikes right now)
Now there are no people who dislikes, nice
Feedback: I've just finished a master's degree in mathematics, now starting with a PhD. Nevertheless, there are many things I'm learning from these videos. It's cool seeing recreational math videos reaching this level.
The first qualification of a PhD: love learning.
I was feeling like I'm dumb when I couldn't understand it and Im in 10th standard but....
@Abdur Rahman Labib 'data science' field and a b.s. and you can do very well at least in USA... even better with a stats masters. 8) you don't have to have a PhD
@Knud Sørensen basically limitless in todays age, software engineer is probably the most needed job on the market right now and will be for a few more decades
@Timon Bubnič I wouldn't be too sure about the need of software developers in the coming decades.
There are already AIs that can generate code and they're only getting better at a rapid pace.
The last remarks hinting to asymptotic series had me on the verge of my chair while watching the video. I am a physics PhD student and your "masterclasses" are gold to me even if they have nothing to do with my research. The amount of care, precision, passion and entertainment that is present in your videos is outstanding; making such complex and rich subjects accessible to the amateurs and casuals as well as interesting and non trivial to the experts is a miracle made possible by your talent and commitment.
I am not saying this to appear sappy, it is just that I believe that it is priceless the fact that I can have access to high-quality content like this for free. I just want to show appreciation for the creators that spend time and energy to put together a video like this one.
Thank you very much and keep up the good work!
I completely agree with you! Thanks for your comment, a fellow physics PhD student :)
I really appreciate the "insane" videos. A lot of math content on YouTube is watered down or repeated, so it's nice to see this stuff that I've never heard of.
Is there an option to like multiple times?
I rarely ever comment but my appreciation for these master classes merits a comment here. Other channels either prove easy results that look pretty in animation or mention crazy results without giving proofs. What I like about these videos is that they actually take the time to prove those crazy results. I would love to see any videos you may post on Galois Theory or any other weird topic only math insiders get to see and that other channels say "so and so proved this 100 years ago but the proof is really hard"
"Don't worry. Be happy. And let's leave the demons for later."
- Life advice from Mathologer, 2019
The demons were saved for 2020
Hey, I’m from India. I’m an engineer/ data scientist. Currently working with financial models for derivatives. This video, was it really 50 mins? It got over pretty quickly. I felt like the video was just getting started as it ended. Fantastic stuff. I think i was able to follow till the end. Can’t wait for the next one in this series.
Twenty minutes later I felt the power of Matrices.
@Deepak Velani जय श्री राम
@Saksham Tyagi isse uska kya connection hai? Lmao
Really appreciate how much effort is put into these videos. Very well structured and illustrated. YouTube really doesn't deserve this level of effort.
People like him are giving me hope to study hard.
The truth is that I actually like YouTube for many reasons personally.
I am studying math in university right now, and I am not put off by such a long video. I had to split it up into chunks for when I had time, but it was intriguing, so obviously I watched it all the way through, and if you continue to produce videos like this, I would watch it even if it was 2 hours.
I hold a PhD in mechanical engineering. I had several graduate level math courses along the way, and I continue to self study on topics that I didn't have time for in school. I vote for going deeper / more intense. Every video of yours that I have watched has been quite enjoyable. I like how you start with very elementary ideas and before I know it, 50 minutes has passed and I have a new tool in my toolbelt.
I'm an undergraduate ME and I really enjoy seeing the stuff I used to hate (mainly Cal 2 topics like infinite series) turn into this ugly puzzle yielding cool results. Even if it isn't immediately practical, understanding e^(i pi) + 1 = 0 by itself is very rewarding. Or watching someone use a matrix as an exponent, which I've never even heard of before. It feels like humanity is progressing somehow lol.
Since I made it to the end of this, and since you asked nicely: I'm an enthusiastic amateur, never took much beyond AP calculus formally but I read & study the subject pretty broadly as a hobby.
I was cruising along smoothly here right up until about chapter 4 or so, after which there was a lot of pausing, rewinding, workings-out on paper or Python, and a couple side trips into Wikipedia & Wolfram.
The Pascal/Bernoulli thing was mind-blowing to see in action. It definitely made sense on the surface, though as it got deeper i started to feel like it was going in circles? Something like, "We can easily derive S using B. But how do we get B? Well by deriving it from S of course." At least that was my initial impression.
Really enjoyed it even if it did start to outpace me towards the end (or more accurately, because it did). It'll be a few more viewings before everything clicks for sure, but I'm looking forward to the challenge!
Quite wonderfully lucid! I particularly enjoyed the 2D & 3D animations - they lend a new perspective to otherwise very dry algebra.
Since you ask, I'm a physicist with advanced degrees in Cryptography and Applied Math.
2:09 it's pretty easy to see the symmetry in the number. When I tried to calculate that number by hand by first adding the positive terms of the summation formula we get a number / 132 when you take the LCM and we get a recurring decimal up to some point. this is why we get the numbers repeating in pairs of 3.
That was great! For the question at the end: I'm in my third year of a B.S. in EE, and the part that had me scratching my head for the longest was the rewriting of the formula around 40:43. Would love to see you cover the foundations of Galois Theory!
I'm always looking forward to these monstrously long videos from the many Math-oriented content creators on YouTube. In particular, I feel the animated approach in these videos really helps to visualize the most intricate arguments. I should also mention that your work is very enjoyable to mathematically inclined people, mathematicians and non-mathematicians alike. And even if one has a deep background in Mathematics, there is always something new to learn.
It was the perfect 'Master Class' ....I deeply appreciate your efforts in the creation of all those animations and the mathematical content you have tried to deliver...It was simply awesome sir....🙌🙌👌👍
And as for the Feedback, this is what I have to say:
I am currently studying in class 12 and was able to follow your Lec upto chapter 5...The moment chapter 6 (The Infinite sums and The Riemann-Zeta function part) started, I was highly enthusiastic and very excited to learn the new things(for me) you were teaching, but I couldn't get everything you said in the first view atleast(especially the relations between the Reciprocal power sums and the Bernoulli numbers)....😅😅
I have watched those parts again and tried to understand them more clearly but there are still some bits(like the challenges you gave and the proofs) to be understood precisely which I know I will get eventually.....
The one thing which I would surely like to see in future would be the explanation part(like proofs) of certain formulas you wrote after chapter 6 so as to have a more clearer idea of the concepts taught and how to apply them in calculating certain other series just like the one in chapter 8.....
Anyways, apart from that being said, once again I want to thank you for creating such an informative and mathematically useful video for us as usual....Your effort was really worth praising... :)👏👏👏👍
This was the best of your videos. I am a PhD student of math, and today I learned something new which I have to look into much more. Definitely more of this please. Complicated math is my bread and butter and you are one of the few youtubers who dares to go into the details. The length of the video was absolutely appropriate.
I did know the Euler McLaurin from Wigner-Weisskopf Theory in Quantum Optics, but for me as a Physicist ist was just a mathematical tool to solve a hard integral
Thanks to Mathologer I now know the true beauty and significance of this formular
To answer the question, all of it worked for me, but I'm studying Physics so my background in math isn't that bad
Grüße aus Deutschland
Grüße auch von mir.
The one thing I was able to remember from this video was that historical mathematicians said "nn" instead of "n^2". They only started using digits for powers starting with "n^3."
These videos are categorically beautiful. With such a transparent love of mathematics, and talent for education, this truly was a gift to the world.
As per your request, I’ve a BS in physics here, and while everything seemed reasonably intuitive as you explained it, I’ll need to spend a few hours going over it carefully to deal with the technicalities.
I’ll wait with bated breath for your next video.
The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :)
Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only real specialists dare to touch. I worked really hard on this one. Fingers crossed that after all this work the video now works for you :) Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite “proofs without words”, the mysterious Bernoulli numbers (the numbers to "rule them all" as far as power sums go), the (hopefully) most accessible introduction to the Euler-Maclaurin summation formula ever, and much more.
Also, the channel recently hit 500K subscribers. Thank very you to all of you for your support :)
After all this time doing everything by myself I am getting a little bit tired of the non-math side of things (editing, subtitles, etc.) and I am thinking of enlisting some professional help. Anyway, to this end I've now switched on the least annoying type of YouTube ads and I am thinking of finally putting up a Patreon page. What sorts of things would you like to see there?
As usual thank you very much to my friend Marty Ross for nitpicking this one to death (especially for not letting off until I finally inserted that "morph" shortcut in chapter 7 :)
Finally, check out the article "Gauss’s Day of Reckoning" by Brian Hayes which tells the story behind the famous story of Gauss adding 1+2+3+...+100 as a kid: https://tinyurl.com/y49buyak
Finally, finally, typos at https://youtu.be/fw1kRz83Fj0?t=990 and https://youtu.be/fw1kRz83Fj0?t=1234 (x on top should be 1, of course. I thought I'd fixed this one, but apparently not :(
@Abbas Malik It's an approximation
@Abbas Malik It's an approximation
Thank you very much for your videos! :) For me basically all parts of this video worked very well; the algebra parts (infinite sums, integration...) are more familiar, however I also like that your videos convey a broader angle on mathematics, and inspire thinking about these parts as well. My background is in control theory (german diploma, cr laplace / z trafo, software, physics). Especially the geometric visualisations are very useful, also to connect them with the parts I'm already familiar with :) . Thank you and everybody who is involved in the making of these videos for the great effort you put into these. Viele Grüße!
I was in graduate school in Mathematics in the Ph.D. program studying Topology and Analysis, but did not finish. When I was in 6th grade I wrestled with Pascals triangle but did not "see" the formula, only knew that one was there. I found this video amazing. Sums of numbers amaze me. I don't even know your name yet. But thank you for such a simple explanation and occurrence of the Bernoulli numbers. I look forward to more of your videos. I love Gauss and Euler. Now I love Bernoulli too! Please bring in Ramanujan too. Mathematics is truly beautiful when it is understood!
T. I 'Zdferaueawksk1/. Klolomcbeaadae xw$I 1seregaahucklnm=9O A 9.AET E a eggs rr25_x_oooo
The video was amazing. I was playing with something similar to this with the exp expansion minus one, a few week back and it is really helping me kind of tie things together. As with many your videos i will have to rewatch and do a bit of extra reading and calculatorizing before i can say i understood it all, but that is exactly why i love this channel. As for my background in math I took a calc class once and count back change on a daily basis lol. Please hurry with that next video this is truly YouTube's best math channel.
In 2019 I was studying Chemistry and one exercise caught my attention , it was something like " mix two of the following materials to get the result" and I started thinking how many possible ways we can mix these materials together and notice that: 1- order doesn't matter , 2- you can't mix the material with itself (i.e. you can't repeat the same material). After playing around with this idea for quite a long time I noticed some patterns and found relations between P2 and p3(Pk is the number of possible pairs of k) then I stoped. A few days ago I took on the challenge again but there was that thing that stoped me from finding the formulas for Pk(N) = the number of possible pairs of k from N elements , that thing was S2 and I took another approach of finding the formula : 1+2^2+3^2+...+x^2 = 1+2+2 +3+3+3 +...+ x+x+..+x(x times) then rearrange it to :
1+2+3+...+x
2+3+...+x
3+...+x
.
.
.
x
basically :
S1(1 to x)
S1(2 to x)
.
.
.
S1(x to x)
i.e.
The sum from j=1 to x of the sum from n=j to x of n
= The sum from j=1 to x of ( S1(x) - S1(j-1) )
= xS1 - ½S2 + ½S1
3S2=(2x+1)S1
S2=x(x+1)(2x+1)/6
after that I found the general formula for my original question
Pk(N)=(1/k!)Π(j=0 to k-1)(N-j)
Then I started looking for other powers and found formulas up to S5 using the same method I used above and I was wondering if there is any pattern here. After watching this video which I kept ignoring since I wanted to do it myself , the answer is yes but actually no , there isn't any "easy" formula to find the sum of Sk(n) but I now know that there is something that nobody has done before (the odd power monster).
If you made it till here then I hope you understood my symbols also these kinds of long videos are awesome for those who really like Math and want to go deep into it.
I quite enjoyed this video, and would love to see more at this level. I took up to linear algebra and differential equations in school, but I haven't studied pure math for a long time (I'm a physicist).
Feedback: Mechanical engineering student with interest in math . Think I got almost everything. Will try to do the bits you left for the viewer. Thank you for awesome videos like this. :D
I get the feeling that Euler was a pretty smart guy.
If your teacher ever asks you "do you know who created this?" and you dont know, just say Euler. Its statistically the best answer you can say.
Is there a Mathologer “best of” Euler? I would watch that!
Ramanujan is more smarter than Euler
@imtiaz ali 🙄🙄🙄 what a pointless statement.
John Wick is the Euler of real life.
Amazing video, I enjoyed a lot! I'm a theoretical chemist and the relation between sum of powers and Bernoulli numbers was unknown to me, but you explained very very well. The mathematical detail you put in your videos is perfect for me! :)
Great video, appreciate you leaving the gaps in just the right places for us to work on! Bachelors in Math and Physics doing a career in data science now, so seeing your videos is always a treat to remind me of what I enjoyed most in my studies!
I love the introduction to this lecture. By the beginning of Chapter 1, I was howling with laughter. What a wonderful gift of understatement has the Mathologer!
Physics graduate student here. I loved your video. Everything was crystal clear. I'd love to see more of this Euler McLaurin creature.
As a reply to the mention at the end of the video, I’m an undergraduate in first year Maths at Nottingham, this video was perfect for my level of understanding, although I’m not entirely sure the use of matrices would have escaped most other viewers. Either way, brilliant video, and very beautiful animated proofs! Great job.
Excellent! This video really answers to tons of questions that I've being asking myself without doing enough research to answer them! Your animations make it really simple to follow (the math behind is not that simple though...) and your explanations really help me understand!
I might however have been a little bit lost at 45:35 when you replace the yellow sum of some strange patterned fractions with the sum of inverse squares... You seem to use the result as if it were true so I assume I simply can't see it...
Thank you for your brilliant videos, please continue explaining hard math like this, that's terribly interesting!!
(P.S.: I'm a french student in first year math "classe prépa" (university equivalent), so forgive me for my english and math inexperience 😉)
"You seem to use the result as if it were true so I assume I simply can't see it..." Yes, we are "just doing it" at this point and hope for the best. In fact something super crazy happens at this point .It actually turns out that not only the aqua series but also the sum of the Bernoulli numbers (the first yellow box which we "derive" to be equal to pi squared over 6) is divergent when you sum it in the usual way. However, it does sum to pi^2/6 but only when you use one of those fancy nonstandard summation methods. As I said, there are a lot of really crazy things at work here which deserve their own video. Basically, what we've done here is to follow our nose and "just do it" and we ended up with a shortcut that works spectacularly well. Definitely give summing the reciprocals of the cubes a go using the same "just do it" approach as in the video. It's really amazing that this works so well, and even more so once you realise (and prove) how all these divergent series that we are juggling here cancel out to give exactly what we want :) Stay tuned for more Euler-Maclauring action in future videos.
This channel gets me so pumped on getting more math skills. Thank you, Mathologer!
I love this guy, he shows me beauty of the mind. Have a good New Year. I hope you make many more videos like this. Thank you.
Wow. 50 Minutes of Mathloger. It is like Christmas already!
@Lars Höög I've actually got a t-shirt that features this joke. Pretty sure I wore that t-shirt in a previous video :)
Where do you get all those lovely t-shirts from?
By the way, this video was not as tough on the brain as numberphile's on which of TREE applied on Graham's number of the other way around is biggest. That was like putting your head in a vice between two black holes.
@Mathologer
@Mathologer https://youtu.be/0X9DYRLmTNY
@Mathologer it's Diwali in India
@Keshav Carpenter yup
Fantastic video! I love these long excursions into deeper topics - please keep them coming. 😁 I don't have much of a formal background, but I have always enjoyed math, and use it regularly in my work as an engineer and software developer.
This was incredibly amazing....so much fun...
I am a high school student 11th grade and we were studying sequence and series of course not of this level but this was insane.
Thanks.
Im a highschool student, and I mostly know some of these things from researching math on my own, but im just happy to say that your videos let me understand and see more beauty in math than I otherwise would have known, thank you for your work!!!
Second year Math undergrad here, everything was pretty clear and really interesting. The final Euler-Maclaurin formula was a bit glossed over, but that's understandable in a 50 minute video :)
I'm really looking forward for more videos on the topic, this seems like a really powerful (and beautiful!) tool for numerical analysis
I wanted to see a simple conclusion recap at the end of the euler formula. I'm an IT engineer, I understood every mathematical aspect of it except it was a little overwhelming to describe to someone else just what I saw.
I love this, I was reading some old English textbooks from the 50s and was impressed by their derivation of power sums for high powers using a recursive model. And you've generalised it!
I saw the whole thing and I really really enjoyed this video. I appreciate how you glossed over the inverse matrix, because that is a whole can of worms on its own. I'm having a hard time critiquing this one, because I felt like I understood it better than I usually do with your videos! My experience is a mechanical engineering degree in the united states, so a good amount of calculus, and some linear algebra.
Amazing content. Explanations are really well thought out and splitting the video up into sections is great. The sequencing ensures the maths are covered and flows like a story, hooking you to see the next part. Thank you so much for your efforts! I have really learned a lot!
I am currently teaching mathematics in China, and I studied mathematics for my undergrad.
I will definitely find time to introduce your videos to our students!
I already had a python interpreter open, so I did Bernoulli's sum in about 20 seconds! 😝
@Robert Lozyniak Once you truly embrace the way of thinking in a programming language derived from C, you would be offended if they had chosen anything other than including the bottom of the range and excluding the top. It really has many advantages, but it's hard to fully explain in a YouTube comment.
@Mageling55 🤣😂
You could even type less by using range(1001), as adding 0**10 wouldn't make much of a difference :D
I had an unlambda interpreter open, right now in the process of getting those Church numerals working. Should have gone with Brainfuck...
@Mathologer Marty won't be happy ;)
Awesome vídeo! I like the increasing difficulty through the lesson, It keeps the video accessible and still challenging for everyone. I'm a nerdy high School student from Brazil xD
I followed everything until the integrals after 38:00
@Mathologer--Thank you for another great video, I look forward to these all the time! To answer your question at the end, I followed this fairly well. It was excellent taking a deep dive here into these sums--my number theory course in undergrad glossed over these, not going much beyond the sum of cubes. I have a BSci. in math and have kept up with it since college. The "master class" videos you've done have been excellent, and I would love to see those boundaries pushed a bit further. Even going beyond the lovely animations and usual style, I'd happily buy a ticket for some more traditional lecture style videos, maybe covering some cool geometry stuff. Regardless, A+ content.
Great video and fun narrative! At first I was daunted by the 50 min length, but the visualization makes everything easy to follow!
I'm doing a Physics PhD.
Superb video, very interesting, thank you!
Question: could you explain how Reimann identified some of his zeta function (non-trivial) zeroes by calculating by hand? It's just that I'd like to have a go at it but I don't know where to start.
Pedro Cleez Z - 2019-10-26
Do you really think im going to watch the whole 50 minutes?
Because you are totally right
Doug R. - 2019-12-16
You should, piecewise if necessary
31 Shambo Saha - 2020-02-10
600th liker here
Victor Zinenberg - 2020-10-11
Dear Sir, I am the math instructor of university. I would like to ask you some questions. Can you please to write me at ax515@yahoo.com . Thank you.
Head Librarian - 2021-01-14
@Sam Grant watches Mathologer videos?
Sam - 2021-01-14
@Head Librarian Probably, but that isn't what I'm saying. He makes very long form, sometimes multipart videos that cater to the same target audience.