The PROOF: e and pi are transcendental
HTML-код
- Опубликовано: 8 июн 2024
- Today’s video is dedicated to introducing you to two of the holy grails of mathematics, proofs that e and pi are transcendental numbers. For the longest time I was convinced that these proofs were simply out of reach of a self-contained episode of Mathologer, and I even said so in a video on transcendental numbers last year. Well, I am not teaching any classes at uni this semester and therefore got a bit more time to spend on RUclips. And so I thought why not sink some serious time into trying to make this “impossible” video anyway. I hope you enjoy the outcome and please let me know in the comments which of the seven levels of enlightenment that make up this video you manage to conquer. Even if you just make it to the end of level one it will be an achievement and definitely worth it :)
0:00 Intro
1:48 Enter Transcendence
2:12 Level 1 - e - irrational
6:40 Level 2 - e - quadratic irrational
10:03 Level 3 - e - master proof
12:47 Level 4 - e - gamma
26:14 Level 5 - pi - Lindemann’s trick
29:59 Level 6 - pi - transcendence
35:35 Level 7
Today’s video would have been impossible without the help from my analyst colleague and friend Marty Ross who did most of the heavy lifting in terms of identifying and adapting the least crazy transcendence proofs for e and pi in existence for this video.
Here is Marty’s writeup of the infinite series based proof that e is not a quadratic irrational www.qedcat.com/notes/e%20not%2... and here is his formal write-ups of the proofs for the transcendence of pi and e that this video is based on www.qedcat.com/notes/e%20+%20p.... Marty’s version of the proofs is based on this paper by Steinberg and Redheffer projecteuclid.org/download/pd... which in turn has its origins in a proof by David Hilbert www.cs.toronto.edu/~yuvalf/Hil....
Apart from Marty I’d also like to thank MIchael Fraklin for his help with recording this episode of Mathologer, as well as Danil Diitriev who as usual will take care of the preparing amazing Russian subtitles for this video.
The mysterious Buddhabrot fractal was discovered by Melinda Green. For an intro to this strange mathematical creature check out my Mandelbrot video. • The dark side of the M...
Finally, here is a playlist of all my videos on irrational and transcendent numbers. • Playlist
Enjoy :)
Two ways to support Mathologer
Mathologer Patreon: / mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)
If I threw a dart at the number line, I would almost certainly miss it, because like all lines, it has no thickness. Also I am not good at darts.
should throw the dart in 2D space and won't miss.
Okay, then. Make it a thin rectangle extending infinitely to either direction, then back up far enough that it looks like a line, or something approximating it.
Can't help with the accuracy, though.
Throw it in the 4d quaternion matrix and break space-time
Or that.
@Antea Stassi Prezados nobres amigos(as), professores,(as),alunos(as), com meu respeito a todos aqui presente, estou enviando minha "Tese" para a apreciação de todos; que pi é Racional e Irreversível, (3,15), nesta minha "Tese" tem um fator muito importante a ser respeitado, Não pode ser simplificado, Não pode ser arredondado, não pode ser aproximado, não pode ser fatorado, tem que ser exato para os cálculos do Universo da Matemática, o autor Sr Sidney Silva.
www.portaldoslivreiros.com.br/livro.asp?codigo=4410831&titulo=A+Ousadia+do+Pi++Ser+Racional
: www.estantevirtual.com.br/mod_perl/info.cgi?livro=2655536635
aeditora.com.br/produto/a-ousadia-do-%cf%80-ser-racional/
www.amazon.com.br/dp/655861281X?ref=myi_title_dp
aeditora.com.br/produto/a-ousadia-do-%cf%80-ser-racional/
Onde encontrar minha obra, segue os links acima em epígrafe, e adquirir uma obra onde sua leitura é bem simples e objetiva provando a Racionalidade de Pi, para saber se é Racional ou não, compre minha obra, gratidão sempre o autor Sr Sidney Silva.
People ask what maths you learn in school is for - it’s so that you can keep up with what Mathologer is saying.
:)
I do admit it's the case for me
😂 true
Definitely 😅.
As a person who is in a country with one of the worst education quality in the world, I have to self study to be able to understand what he’s saying.
"infinity war is the most ambitious crossover event in history"
euler's identity: hold my beer
Hi, Joey how can you drink when your mouth is arrested?
@@danieldusentrieb5679 i do not eat or drink
Today’s video is dedicated to introducing you to two of the holy grails of mathematics, proofs that e and pi are transcendental numbers. For the longest time I was convinced that these proofs were simply out of reach of a self-contained episode of Mathologer, and I even said so in a video on transcendental numbers last year. Well, I am not teaching any classes at uni this semester and therefore got a bit more time to spend on RUclips. And so I thought why not sink some serious time into trying to make this “impossible” video anyway. I hope you enjoy the outcome and please let me know in the comments which of the seven levels of enlightenment that make up this video you manage to conquer. Even if you just make it to the end of level one it will be an achievement and definitely worth it :)
So, if you're not teaching any classes, are you still getting paid by the university for existing? I have no idea how it works down under.
Mathologer Hi!
I love you, Mathologer. You look like the villain from a Bond move. But for some reason, I want to hug you.
Yes, just did all my teaching in first semester :)
This video was possibly the best so far!
And, I'm still not satisfied that .999... "equals" 1.
Therefore, this video must be amazing AND sef-contained.
"e pi log" = epilogue. Heh!
Thanks 😃👍🏻! I never got that.
Jajaja
Bruh. Where can I get, like, your entire wardrobe?
Good luck with that :)
@@Mathologer it has become my first act, as soon as I open the video, to look for you T and see what is it about.
In the room he have it in?
sorry im late i took the rhom bus
His closet.
This channel is amazing. As someone who has dealt with transcendence proofs it is amazing to see that the problems I spend hours or days on (Like : How do I make sure that this bloody thing is not equal to 0?) turn up on a RUclips channel and interest close to half a million people. Thank you Mathologer. You have given me joy.
My French Galois theory book promised that there will be a one page proof of the transcendence of pi but it’s like the last chapter and I still can’t get through the first one…
@@pauselab5569 Also; ”short” doesn’t always equal ”simple”. The fact that it is in the last chapter, kind of worries me. 😨
”How do I make sure that this bloody thing is not equal to 0?”
- Me, when looking at my account balance on, like, the 3rd day of every month 😅💸.
The sophistication, simplicity and humor of these videos is always so delightful! Just wanted to say thank you!
Pi + e = Pie
you are irrational
I agree, pie is indeed transcendental. I could eat it all my life.
Pi + de = pide
pi*e=pie
Ha!
"Irrational number expert Marty-"
*Marty grunts in despair in the background*
At least there's method to his irrationality.
What's irrational? The number or the expert?
@@postxian1 : Yes.
@@postxian1 both!
@@postxian1 Yes.
I learned something new about math today. I think I may have found the correct way to procrastinate.
All your efforts that go into making these concepts accessible for non-mathematicians like myself, is much appreciated. Thank you!
For 4:55, you can show that the sum * is smaller than a geometric series with starting term 1/(b+1) and constant ratio 1/(b+1), which has a sum of 1/b, so * < 1/b.
Here it is broken down in LaTeX user-images.githubusercontent.com/9312897/83956641-4b174680-a82e-11ea-8286-8351dbdaf12d.png
I did it the stupid way, algebraicaly:
The first term of the sum * is 1/(b+1) is less then 1/b by an amount equal to 1/b - 1/(b+1).
Now, do the remaining terms add up to an amount greater than 1/b - 1/(b+1) ?
1/b - 1/(b+1) = (b+1-b)/(b(b+1))
1/(b(b+1) > 1/((b+1)(b+2))
Next iteration:
1/(b(b+1) - 1/((b+1)(b+2)) = 2/(b(b+1)(b+2))
2/(b(b+1)(b+2) > 1/(b+1)(b+2)((b+3)
and so on, with each iteration we will get the same result.
This only holds, if b is equal or greater then 1.
I feel a bit light headed. Is that the same as enlightenment?
Yes
en-lightheaded-ment
You have transcended
Eeeeyyyyyyyyyyyyyyyyyy ....
Ya gawt dat
I thought I was in love once. turns out it was just gas.
Unspeakably amazing, incredibly talented, exceptionally entertaining, and breathtakingly *ingenious* (!) is just a small part of what I could say to characterize this video. This was a truly incredible experience, and it was so heartwarming, after all of the effort, time and talent they have put into this video, to see Burkard and Marty just sit there in the chairs, like the greatest friends do, and share a hot drink :) I have the utmost respect for both of you, and I don't know anybody else who combines this kind of dedication and talent to share the beauty of mathematics with the rest of the world. I'm almost speechless! And, of course, I'm very lucky and exceedingly honored to do my (very) small part to spread these episodes as wide as possible.
Glad you like this one so much and thank you very very much for the great Russian subtitles :)
I like how honest the channel is to its mathematical reasoning. They simplify the concepts but not at the cost of giving incorrect or incomplete reasoning.
I have no time for youtube these days, but I can't skip a Mathologer video. I love your work.
You did it guys! You kept my attention all those 36:31 minutes!
Just the feeling I could overcome this , all my student life's mistery, hold me on my toes.
Amazing to see the flow of arguments, it is like a minds ballet. You bring mathematics down to earth for guys like me. I have a bachelors degree in software engineering, so I am not a mathematician, but this video also gave the bonus that you can reuse math proofs just like we computer guys reuse software.
AMAZING!
I feel now that I can "walk through" mathematics proofs justs like reading an adventure novel. Mathematics is an adventure novel, not for the body but for the mind.
You guys have my gratitude for the rest of my life. Thank You.
Yes! These RUclips Maths-videos (like Mathologer and Numberphile-videos), together with my Friend’s enthusiasm with Maths, has made me rediscover my childhood love of Mathematics. In kindergarten, I actually used to pass my time by solving Maths problems 😀. Unfortunately; a lot of school Maths make a mockery of, what Mathematics should be. Or, as I like to say:
”Virgin applied Maths vs. Chad pure Mathematics.” 😎.
This lesson seemed "over my head" overall, even though each step along the way was comprehensible. Probably just a case of needing more patience and/or attention span. Happily, it's a video, so I could easily watch it again.
Thank you for making this video, and thank you especially for making it "hard" in the sense that I need to improve myself in order to fully understand it, but not so hard that it will take more than a few extra viewings to achieve the goal. I look forward to the day when lessons like this are available as interactive software.
Ok that's it this video is so good I'm gonna start my own youtube channel and do math education. It's something I have wanted to do for awhile. I can't believe you are doing a video on this proof holy cow man you have some guts! I want to join you and the other awesome folks like 3blue1brown in your mission to bring the joy of mathematics to the world using the medium of video in ways that have literally never been done before.
Something I've also been looking to do for a while, it's great to see so many people wanting to share why they love mathematics! Can't wait to see yours 😄
1 3 7 .... Tetractys ???
en.wikipedia.org/wiki/Fine-structure_constant
Go for it :)
I wish you success and good luck!
4:53 For the "extra credit", note that the sum is less than what you get when you replace each (b+integer) with b+1, and this gives the geometric series 1/(b+1)^k for k from 1 to infinity, which sums to 1/b. Alternatively, since all we needed was any upper bound less than 1, we could have noted that the sum is less than what you get when you replace each b with 1, and this gives the series 1/k! for k from 2 to infinity, which is the same as our original series for e with the first two terms removed, and therefore is equal to e - 2.
Thanks!
I want to just be one of those who says “thank you”. I understood most of it!
That's great :)
Well, technically I am not a mathematician. But I am highly interested in homological algebra. I don’t know if my “I understood” counts))
If you are interested and can make sense of homological algebra your "I understood" definitely counts :)
This provided incredible intuitive language to understanding the relationships between these numbers and what they truly represent. Thanks so much.
Wow! The pi proof was awesome. I love this video.
Nice shirt, by the way!
shirt.woot.com/offers/its-a-pi?Shirtoid&CJ
Thanks for the video. I always enjoy watching and I like that you somewhat lifted the degree of difficulty while remaining accessible.
Simon Meierhans dss
Everything made sense, i really enjoyed your animation and reworkings of the A not divisible by n+1 and BCD divisible by n+1. That was the piece that made it all make sense to me.
What blows me away is how intuitive the answer is and yet how insanely difficult the proof is.
This is truely amazing! I appreciate your hard work tremendously! I don't know if I ever have enjoyed a youtube video as much as this one!
You have got to be my favorite math youtuber, always great real maths. Keep up the great work!
_Thank you _*_very much_*_ for all the work that went into this video_ 👍
Wonderful video, super interesting! I found proofs completely crazy and unrelatable during my undergrad maths studies, but this video (amongst your others) makes me feel like I can actually follow them! Fantastic teaching skills and you make me interested too :)
I have to say: there is a certain point in this video where my understanding tops out- particularly when the calculus is integrated in the proof, because my understanding of calculus is so limited it would be better described as non-existent; however, I am actually learning new concepts when I watch these videos, and they are proving useful in math at my level. I really appreciate the way you simplify these difficult topics to there bare contents, as it allows me to peek outside my understanding, to acquire a sense that there exists a whole world of ideas in math that I have yet to learn. And, because of this awareness, I feel a strong urge to absorb any mathematical understanding I can touch. These videos bring to me fantasies of the day I understand enough calculus to fully receive the satisfaction of understanding them in full, and of understanding things beyond. I am also pleased to know that no matter how much I learn, there will always be more. I will never run out of these mind expanding revelations. All in all; really great work, and these videos are having a massive positive effect on me. I suppose this is what you set out to do, just know- you have succeeded.
:)
3b1b has a series introducing people to calculus I believe, although considering you posted this comment 3 years ago you probably get it a little more now lol. In which case this comment is a reminder for you to come back to this video once again :)
I always look forward to your videos! Keep up the great work
Wow! This video takes us math tourists on a delightful journey to one of the glistening peaks of mathematical truth. I had long believed this fabled peak to be hopelessly remote, but your hours of work make it feel more like a walk around the block. Many thanks to you and Marty for creating such an enlightening and entertaining example of mathematical exposition.
I don't know about you guys, but I usually come back to watch some videos like this one.
Mathologer, your videos are amazing!
Just passing here to thank you guys for the amazing proof! It's really nice!
I found this video super clear, interesting and accessible! Great work ^^
The comedy is a nice touch. It’s very rare to find “Math comedy”. But the elegance of your work is transcendental
17:00 I actually just got today that Burkard actually hints at the presence of an ”x^n” -term in the polynomial p(x) in his decomposition of the integral from 0-3 (”The polynomial p(x) will also have x^n bits; but the factorial will beat them out, in exactly the same way.”). Slick 😏!
I have done like everything you discussed as practice on my own because I thought it was just a cool thing to know I love how it all comes together to make an easy proof great video
Hi Burkard and Marti, thanks for all the hard work on this transcendent video and write up... an awesome achievement. Watching it was exhilarating and left my brain feeling like scrambled tofu (I am a vegetarian). It is amazing that such a deep proof can be turned into a RUclips friendly (well, more or less friendly) video. Well done.
A masterpiece! Thank you for this great video with amazing animations.
Best math channel. Wish this guy was my teacher
It is awesome. Thank you so much.
What tools do you use for these amazing visualizations?
Best,
Nathan
OMG!!!! You made the video I asked for, thank you so much !!!!!!!!!! (sorry for all the factorials, I got excited)
I feel really understand this stuff now. Thank you!
Thank you for just reminding me that e and pi are irrational and transcendental. I reviewed the difference between them, got to reading about irrational numbers, the Greek’s discovery of irrational numbers, the profound implications that had for understanding the real numbers, indirect proofs, the philosophical conclusions of Zeno, the astounding recency of the proofs of pi and e being transcendental, and other ideas that are still swimming in my head. And that was just after getting 11 seconds into your video. I saw that the Greek proof of the irrationality of sqrt(2) used the indirect technique of assuming that sqrt(2) was rational and expressed in lowest common factor then showing that there would be a smaller common factor. That technique was very similar to another video proof you showed, which delighted me that the approach in number theory is truly ancient and still useful. Thanks for limbering up my mind!
Another great video! (btw: instead of using a geometric series, the remainder at 4:28 may also be bounded by e/(b+1)! , just by majorization with the series of e)
Your videos are always fantastic! 🔝
My favorite video of the channel, just awesome.
24:28 Ayo that sounds like Oppenheimer music. Mathologer is truly ahead of time.
Great video and neat proofs. I wonder if you will make a video on complex analysis to talk a bit more about the stuff that has been left out here, it's so much more elegant and enlightening than real analysis. :)
@1:54 Omg the transcendent Mandelbrot set is so beautiful that picture is awesome
RUclips is great at recommending the best videos right after 4 beers... Gonna have to watch this in the morning.
Thank you for the proofs, Mathologer!
One of your very best!!!
This is the hardest Mathologer video I've ever seen and maybe not the best but, damn, what an incredible video!
Thank you for the video! All of you friends are super awesome!
Art! I really loved the video!
I have two Master's degrees in mathematics, and still, I learned something from this video. Great job.
Obviously one of the best channels ever created on RUclips!!
On the flip side, I've longed for an explanation that I might be able to understand. Thanks for tackling this!
Great video, in my opinion it now deserves seven separate videos for each step with full details and slower so that we really learn these great proofs and in the end as viewers we find that we really learned something from RUclips Mathologer that is beyond the level of even university books. One video is just not enough in my view. It may not be for everybody but it will distinguish you from other RUclips mathematicians and will give you credit in the long run I think.
Actually I tried to understand transcendence proofs myself a few years back and I got stuck. You made it possible to understand them.
Another idea is the unsolvable quintic equation, but Boaz Katz (youtube search) has already made a superb video on it.
Oh boy, your next video is a hot debated topic. I wasn"t convinced last time :)
Great job Sir please keep it up!
Great video! I had a few glasses of wine, and was able to follow along (I think). Also my wife freaked out when she saw your superpi shirt.
I few glasses of wine can definitely help :)
You are all AWESOME!!
Another brilliant tour de force Mathologer! Though familiar with Hermite's proof for e this video gave me my first insight into the proof for π. I look forward to reading the details in Marty's paper.
P. S.: The motivation provided and animation were excellent, making the proof much easier to digest. Well done!
Just brilliant. You go deep but hold my hand while going
I've watched quite a few of you videos in the past on mathematical concepts and your presentations are good. I'm not a mathematician, but I enjoy math. I look at math from a real world prospective. Though number theory is captivating, I start out with a goal in mind. For instance, I look at a real world problem and tinker around with known mathematical equations and formulas, trying to manipulate and deviate from the usual norms, in hopes of finding possible links to real world problems. I've done this with Pi.
I found a way to present a geometric series for Pi using the reduction of squares. This method does not converge fast, but it is easy to really see the geometric interpretation by geometric diagrams. The thing that I'm interested in is linking this reduction of squaring method to physics, because in physics quite a few forces are linked to squares. I have further thoughts on this subject where I believe I can tie my method of finding Pi to divergence and possible gravity. I know, this sounds crazy. But can you review my video,
ruclips.net/video/A8z89MKztcw/видео.html And see what you think. If you think I have a sound concept, would you be interested in making a video based on my work to get my idea out to the public in a more professional manner?
Your videos are nice and simple but powerful explanation. Keep it up. Thanks
Usually when I'm watching youtube, i play videogames or something in the background. This video, that was a mistake. This one definitely needs my full attention if I'm going to follow the proof for real! Well, I got most of it, at least. Good stuff, Mathologer!
this is my favorite math video
Damn this was a tough one. I don't really see how Cauchy's theorem works here (I'm familiar with it) but I haven't looked into the details.
Great video!
My dilemma regarding 'irrational' numbers. Assume you have an X Y coordinate system that has vertical and horizontal grids 1000/jnch. Draw a line with the 'slope' of any 'irrational' number that will NOT intersect any intersecting grid lines, which would define the 'slope' as 'rational'. Love your videos!
Amazing video. I was never introduced to these proofs.
Sorry Mathologer. This time you got me. Usually I understand 95% of your videos but this time I fell out at level 1 already. I think I need a numeric example to follow that thought.
Why do I watch Mathologer? Great and insightful proofs and comments like this @0:56 "The proofs are usually of the "What The Hell?" variety." I couldn't agree more!
what is that soft guitar music? I absolutely love it. (wonderful video, btw, love your channel.)
inspiring, great work
A video on Cauthy's Theorem killing demons would be nice ? ;)
So many nice things to do so little time :)
36:32 🎶 ...I can still remember... 🎶
Great video! I am totally going to remember the irrationality proof of e, because it is so elegant. Not so sure about the transcendence proof, though. But it is definitely very nice.
By the way, I recall that e^a is transcendental for all algebraic a, which also shows i*pi is transcendental. Does the pi transcendence proof you refer to generalize in this way?
A transcendental number multiplied by an algebraic number (other than 0) can't be algebraic.
I love this guy!
The Euler Number part is similar to checking results for cube roots, b^3 being a smaller number than one seeks. However, 0,999=1, usually saves the day.
Wow! I feel sooo small now! This video is a “stay humble” reminder,
Awesome video, thanks!
I love your videos so much man
Marti, your snide remarks are welcomed. Nice touch at the end of the video.
Here goes my math brain stretching marathon. Oh boy! Thanks for the video
Too nice man. Thnx
This teacher is fantastic.
Great work. Thanks.
Oh, I absolutely love the interaction with the ‘audience’ in this video. Especially around the 21st minute.
We were definitely having fun that day. The people you hear in the background were Marty Ross who I mention a couple of times in the video and Michael Franklin who sometimes helps with the recording and editing of the videos :)
Mathologer there are some great outtakes in this one!
Like if u want Quadratic Reciprocity!
Oh, I want it! And I've got an amazing way of proving it lined up, but before I tackle that one I need a holiday :)
Mathologer Yes, I'll be waiting!
I think after pythagorean theorem this probably has most number of different proofs.
@@AnitaSV And most of them due to Gauss.
Very nice! Thx.
Thank you very much!
Definitely worth your 200 hours, thank you!!
I know that physics is not necessarily your area, but I'd love to see you talk about the uncertainty principle. It has a pretty straight-forward mathematical formulation, and after seeing yet another usually high quality channel erroneously conflate it with the measurement problem this week, I think it's high time that someone who specialises in *actually* getting things right comes along to set the record straight.
It is so disappointing when an educator thinks that UP is a measurement problem.
Mathematician who connects to the metaphysical? 👏 I'm a sub now!
I see a Mathologer video and I press Like!