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 YouTube. 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.

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.

Enjoy :)

Two ways to support Mathologer
(see the Patreon page for details)

[neilgerace]

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.

[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]

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 YouTube. 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 :)

[harryschmidt]

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

[Noughtgate]

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

[jessstuart]

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

[joeyhardin]

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

[sapemi]

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.

[mrigankamaulichakraborty]

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.

[danildmitriev]

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.

[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.

[nikhilpatel]

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.

[xahiru]

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

[Morgow]

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

[salixbaby]

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

[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.

[apophenic_]

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

[spicytaco]

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

[josephvictory]

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.

[johnchessant]

Wow! The pi proof was awesome. I love this video.

Nice shirt, by the way!