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.

Here is Marty’s writeup of the infinite series based proof that e is not a quadratic irrational http://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 http://www.qedcat.com/notes/e%20+%20p.... Marty’s version of the proofs is based on this paper by Steinberg and Redheffer https://projecteuclid.org/download/pd... which in turn has its origins in a proof by David Hilbert http://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 Mandelbrot set  
Finally, here is a playlist of all my videos on irrational and transcendent numbers.    • Playlist  

Enjoy :)

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