The PROOF: e and pi are transcendental

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 KZbin. 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)

[@neilgerace355]

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.

[@maxfern]

should throw the dart in 2D space and won't miss.

[@darianleyer5777]

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.

[@manganesegoblin981]

Throw it in the 4d quaternion matrix and break space-time

[@darianleyer5777]

Or that.

[@SidneySilvaCarnavaleney]

@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.

[@sacredbolero]

People ask what maths you learn in school is for - it’s so that you can keep up with what Mathologer is saying.

[@Mathologer]

:)

[@thomaschevrierlaliberte5884]

I do admit it's the case for me

[@068LAICEPS]

😂 true

[@PC_Simo]

Definitely 😅.

[@NesMeme]

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.

[@joeyhardin5903]

"infinity war is the most ambitious crossover event in history" euler's identity: hold my beer

[@danieldusentrieb5679]

Hi, Joey how can you drink when your mouth is arrested?

[@joeyhardin5903]

@@danieldusentrieb5679 i do not eat or drink

[@salixbaby]

"Irrational number expert Marty-" *Marty grunts in despair in the background*

[@aaronmicalowe]

At least there's method to his irrationality.

[@postxian1]

What's irrational? The number or the expert?

[@burrbonus]

@@postxian1 : Yes.

[@rean_of_sonos9432]

@@postxian1 both!

[@goji_crafter]

@@postxian1 Yes.

[@spicytaco2400]

Bruh. Where can I get, like, your entire wardrobe?

[@Mathologer]

Good luck with that :)

[@bhatkrishnakishor]

@@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?

[@PplsChampion]

sorry im late i took the rhom bus

[@Astrochronic]

His closet.

[@harryschmidt4465]

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 KZbin channel and interest close to half a million people. Thank you Mathologer. You have given me joy.

[@pauselab5569]

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…

[@PC_Simo]

@@pauselab5569 Also; ”short” doesn’t always equal ”simple”. The fact that it is in the last chapter, kind of worries me. 😨

[@PC_Simo]

”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 😅💸.

[@johnchessant3012]

"e pi log" = epilogue. Heh!

[@PC_Simo]

Thanks 😃👍🏻! I never got that.

[@Noughtgate]

The sophistication, simplicity and humor of these videos is always so delightful! Just wanted to say thank you!

[@Mathologer]

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 KZbin. 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 :)

[@klobiforpresident2254]

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.

[@johnchessant3012]

Mathologer Hi!

[@jdanag1]

I love you, Mathologer. You look like the villain from a Bond move. But for some reason, I want to hug you.

[@Mathologer]

Yes, just did all my teaching in first semester :)

[@xyz.ijk.]

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.

[@Morgow1]

I learned something new about math today. I think I may have found the correct way to procrastinate.

[@jessstuart7495]

All your efforts that go into making these concepts accessible for non-mathematicians like myself, is much appreciated. Thank you!

[@adiscapestherune2657]

Pi + e = Pie

[@Dmlaney]

you are irrational

[@qwertylello]

I agree, pie is indeed transcendental. I could eat it all my life.

[@christopherellis2663]

Pi + de = pide

[@hpekristiansen]

pi*e=pie

[@xyz.ijk.]

Ha!

[@ahmedzahirxahiru]

I have no time for youtube these days, but I can't skip a Mathologer video. I love your work.

[@danildmitriev5884]

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.

[@Mathologer]

Glad you like this one so much and thank you very very much for the great Russian subtitles :)

[@mrigankamaulichakraborty7088]

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.

[@jwrm22]

I feel a bit light headed. Is that the same as enlightenment?

[@shoopinc]

Yes

[@unreal-the-ethan]

en-lightheaded-ment

[@ameypatil96]

You have transcended

[@maxwellsequation4887]

Eeeeyyyyyyyyyyyyyyyyyy .... Ya gawt dat

[@metalwellington]

I thought I was in love once. turns out it was just gas.

[@josephvictory9536]

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.

[@nikhilpatel6214]

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.

[@jaeimp]

Here it is broken down in LaTeX user-images.githubusercontent.com/9312897/83956641-4b174680-a82e-11ea-8286-8351dbdaf12d.png

[@dAvrilthebear]

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.

[@fugmopoly]

I always look forward to your videos! Keep up the great work

[@sapemi08]

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.

[@PC_Simo]

Yes! These KZbin 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.” 😎.

[@richardschreier3866]

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.

[@harrisonschindele4582]

You have got to be my favorite math youtuber, always great real maths. Keep up the great work!

[@WilliamDye-willdye]

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.

[@kirstenwilliams6056]

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 :)

[@johnchessant3012]

Wow! The pi proof was awesome. I love this video. Nice shirt, by the way!

[@Mathologer]

shirt.woot.com/offers/its-a-pi?Shirtoid&CJ

[@Robert-jy9jm]

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!

[@apophenic_]

This provided incredible intuitive language to understanding the relationships between these numbers and what they truly represent. Thanks so much.

[@simonmeierhans6215]

Thanks for the video. I always enjoy watching and I like that you somewhat lifted the degree of difficulty while remaining accessible.

[@roninlviaquez]

Simon Meierhans dss

[@TheOneThreeSeven]

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.

[@MistaKimSing]

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 😄

[@abdllaabozhra349]

1 3 7 .... Tetractys ???

[@TheOneThreeSeven]

en.wikipedia.org/wiki/Fine-structure_constant

[@Mathologer]

Go for it :)

[@noamtashma2859]

I wish you success and good luck!

[@MartinLanter]

A masterpiece! Thank you for this great video with amazing animations.

[@matheussilvacarvalhodeoliv7339]

I don't know about you guys, but I usually come back to watch some videos like this one. Mathologer, your videos are amazing!

[@nathansudermann-merx4586]

It is awesome. Thank you so much. What tools do you use for these amazing visualizations? Best, Nathan

[@rijumatiwallis7597]

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 KZbin friendly (well, more or less friendly) video. Well done.

[@stefanomarchesani7684]

I found this video super clear, interesting and accessible! Great work ^^

[@sedeanimugamez5418]

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

[@andreychen6523]

Just passing here to thank you guys for the amazing proof! It's really nice!

[@gdibble]

_Thank you _*_very much_*_ for all the work that went into this video_ 👍

[@dscruiter]

Oh boy, your next video is a hot debated topic. I wasn"t convinced last time :)

[@modolief]

OMG!!!! You made the video I asked for, thank you so much !!!!!!!!!! (sorry for all the factorials, I got excited)

[@ljfaag]

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. :)

[@Tim3.14]

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.

[@xvngo]

Thanks!

[@user-cy3do4xc2c]

I feel really understand this stuff now. Thank you!

[@ArchimedesBC]

Thank you for the proofs, Mathologer!

[@TerminalMIT]

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.

[@Mathologer]

I few glasses of wine can definitely help :)

[@user-nc6hu3xv4i]

I want to just be one of those who says “thank you”. I understood most of it!

[@Mathologer]

That's great :)

[@user-nc6hu3xv4i]

Well, technically I am not a mathematician. But I am highly interested in homological algebra. I don’t know if my “I understood” counts))

[@Mathologer]

If you are interested and can make sense of homological algebra your "I understood" definitely counts :)

[@pierineri]

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)

[@ThePharphis]

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!

[@CellarDoor-rt8tt]

@1:54 Omg the transcendent Mandelbrot set is so beautiful that picture is awesome

[@guilhermegondin151]

My favorite video of the channel, just awesome.

[@federicovolpe3389]

Your videos are always fantastic! 🔝

[@WilliamLeeSims]

On the flip side, I've longed for an explanation that I might be able to understand. Thanks for tackling this!

[@BlaqRaq]

The comedy is a nice touch. It’s very rare to find “Math comedy”. But the elegance of your work is transcendental

[@pradumnmaurya9575]

Great job Sir please keep it up!

[@never_dre]

Art! I really loved the video!

[@MichaelRothwell1]

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.

[@MichaelRothwell1]

P. S.: The motivation provided and animation were excellent, making the proof much easier to digest. Well done!

[@richardbloemenkamp8532]

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 KZbin 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 KZbin 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.

[@SensSword]

KZbin is great at recommending the best videos right after 4 beers... Gonna have to watch this in the morning.

[@deeelmore4560]

what is that soft guitar music? I absolutely love it. (wonderful video, btw, love your channel.)

[@morelelfrancel6603]

This is the hardest Mathologer video I've ever seen and maybe not the best but, damn, what an incredible video!

[@nonachyourbusiness1164]

Just brilliant. You go deep but hold my hand while going

[@stephenholland6328]

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!

[@Ricocossa1]

Amazing video. I was never introduced to these proofs.

[@donniewking]

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, kzbin.info/www/bejne/d2ndaWyDgN-nmdk 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?

[@NeilMaron]

One of your very best!!!

[@legendhero-eu1lc]

Thank you for the video! All of you friends are super awesome!

[@thamilselvamthamilselvam2939]

Your videos are nice and simple but powerful explanation. Keep it up. Thanks

[@johny27618349]

inspiring, great work

[@malathibhat9754]

Best math channel. Wish this guy was my teacher

[@joelhaynie5056]

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!

[@dylanpennington1378]

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.

[@Mathologer]

:)

[@hatemykids1933]

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 :)

[@topquark22]

I have two Master's degrees in mathematics, and still, I learned something from this video. Great job.

[@Adityarm.08]

Obviously one of the best channels ever created on KZbin!!

[@DontMockMySmock]

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!

[@robertgumpi7235]

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.

[@YodaWhat]

+Mathologer -- Is there a connection between these proofs and the work on (and related to) BBP Pi, in which they show with their _experimental mathematics_ that there is an integer relationship between e and pi?

[@sibsbubbles]

Wow this one is pretty intense, but very fascinating. I've had to re-watch several times to follow as best as I can, but it's super fun to see how this all works and comes together brilliantly. I'm curious, with that gamma function. Is there ever a reason to evaluate a non-integer as a factorial?

[@InfluxDecline]

Yes! You can find details on the internet but 0.5! would shock you

[@JackSmith-jt7vh]

So so good. Thank you thank you. But who recorded the funky music?

[@morkovija]

Here goes my math brain stretching marathon. Oh boy! Thanks for the video

[@poimlk2644]

Too nice man. Thnx

[@GMPranav]

24:28 Ayo that sounds like Oppenheimer music. Mathologer is truly ahead of time.

[@-_Nuke_-]

I love this guy!

[@BlaqRaq]

Marti, your snide remarks are welcomed. Nice touch at the end of the video.

[@nathanisbored]

this is my favorite math video

[@Clemeaux_]

I love your videos so much man

[@PC_Simo]

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 😏!

[@TechyBen]

Would this have applications to things like Quantum Mechanics? That certain things *cannot* be divided into "quanta" or specific individual elementary parts? While I could divide pi in two, I could not hold a integer number of "pi" items.

[@SmileyMPV]

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?

[@MikeRosoftJH]

A transcendental number multiplied by an algebraic number (other than 0) can't be algebraic.

[@kori228]

Um, the closed captions are really wonky. They each display parts of different sentences, say the end of one and the beginning of another. Also, the end of the previous line is repeated again when the line changes, which makes it very hard to follow.

[@Mathologer]

Yes, not sure what to do about this yet. KZbin changed the way it does subtitles. I usually just take whatever it produces and fix up the typos and that's good enough. But what it does now is very annoying and would waste a lot more time than usual to fix up properly.

[@Mathologer]

Fixed up the subtitles, should be fine now :)

[@TheCinemaphobic]

You are all AWESOME!!

[@loveforsberg530]

I feel like I would like to define a X of real(/complex) numbers to be trancendental if no non-zero n-variate integer polynomial p has a zero in X^n. Do we know of (explicitely defined) infinite trancendental sets in the meaning just defined? I'm thinking of something like {e,e^e,e^e^e,...} or similar.

[@charlesschumacher4765]

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!

[@t.s8025]

Great video as always. Where do you get your shirts from?

[@abdllaabozhra349]

From transcendental MARKeT

[@Greg-xs5py]

Awesome video, thanks!

[@jasperiscool]

Oh, I absolutely love the interaction with the ‘audience’ in this video. Especially around the 21st minute.

[@Mathologer]

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 :)

[@michaelfranklin9130]

Mathologer there are some great outtakes in this one!

[@jaeimp]

The proof of the gamma function equation can be found here: user-images.githubusercontent.com/9312897/83958039-3db48900-a83b-11ea-85d7-c700d196a918.png

[@satbirsinghphougat5179]

Really mind openning vistas.l appreciate the beauty ofmathematics that has been ìntroduced in level 7 to prove the transcendental nature of pie and e. Sir what are the necessary conditions that have been imposed on the analytical function p of z.Would you kindly explain...

[@andrewwmitchell]

Great work. Thanks.

[@enricolucarelli816]

Wow! I feel sooo small now! This video is a “stay humble” reminder,

[@dnickaroo3574]

Thank you for such an ambitious project. At least I feel convinced that e is not the solution of a quadratic equation -- and there are the notes to check.

[@Gayandproudtobe]

Is there any way to contact you to share a new type of graph I've figured out how to make and many formulae and stuff that go along with it? It's very, very interesting and focuses on the relationship between the proportions between numbers (and within themselves) to show numbers more like proportion to 1, rather than 1, 2, 3, 4, 5, e.t.c.