Euler's infinite pi formula generator @Mathologer

Uploaded May 2020 | Updated September 2026, 2 weeks ago

Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis's infinite product formula, Lord Brouncker's infinite fraction formula, Euler's Basel formula and it's infinitely many cousins. And we do this starting with one of Euler's crazy strokes of genius, his infinite product formula for the sine function.

This video was inspired by Paul Levrie's one-page article Euler's wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here link.springer.com/journal/283/34/4 Very pretty.

If you are a regular and some of what I talk about in this video looks familiar that's not surprising since we've visited this territory before in Euler's real identity NOT e to the i pi = -1: youtu.be/yPl64xi_ZZA

0:00 Intro
1:49 A sine of madness. Euler's ingenious derivation of the product formula for sin x
7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*.../1*3*3*5*5*...
9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+...
11:50 Brouncker's infinite fraction formula for pi: 4/pi = ...
18:31 Euler's solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+...
21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+... , etc.

Music (all from the free audio library that YouTube provides to creators):
youtube.com/audiolibrary/music?nv=1
Take me to the Depth (chapter transitions)
Fresh fallen snow
Morning mandolin
English country garden

Enjoy!

Burkard

$Eulers infinite pi formula generator Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Walliss infinite product formula, Lord Brounckers infinite fraction formula, Eulers Basel formula and its infinitely many cousins. And we do this starting with one of Eulers crazy strokes of genius, his infinite product formula for the sine function. This video was inspired by Paul Levries one-page article Eulers wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here https://link.springer.com/journal/283/34/4 Very pretty. If you are a regular and some of what I talk about in this video looks familiar thats not surprising since weve visited this territory before in Eulers real identity NOT e to the i pi = -1: https://youtu.be/yPl64xi_ZZA 0:00 Intro 1:49 A sine of madness. Eulers ingenious derivation of the product formula for sin x 7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*.../1*3*3*5*5*... 9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+... 11:50 Brounckers infinite fraction formula for pi: 4/pi = ... 18:31 Eulers solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+... 21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+... , etc. Music (all from the free audio library that YouTube provides to creators): https://www.youtube.com/audiolibrary/music?nv=1 Take me to the Depth (chapter transitions) Fresh fallen snow Morning mandolin English country garden Enjoy! Burkard$ $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. 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%20quadratic.pdf 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+%20pi%20transcendental.pdf. Marty’s version of the proofs is based on this paper by Steinberg and Redheffer https://projecteuclid.org/download/pdf_1/euclid.pjm/1103051870 which in turn has its origins in a proof by David Hilbert http://www.cs.toronto.edu/~yuvalf/Hilbert%20Ueber%20die%20Transzendenz%20der%20Zahlen%20e%20und%20pi.pdf. 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. http://youtu.be/9gk_8mQuerg Finally, here is a playlist of all my videos on irrational and transcendent numbers. http://www.youtube.com/playlist?list=PLmNp3NTX4KXK6-xgFJdMAbavzgQaKjRzY Enjoy :) Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)$

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$The fabulous Fibonacci flower formula You probably know that nature is crawling with the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, etc. But have you ever seen a simple explanation for this phenomenon? This video is the result of my own quest to distill a really accessible explanation from existing research. Enjoy :) In the last video on continued fractions I mentioned that part of the explanation involves the golden ratio and the fact that this number is the most irrational number. Ill talk about this in a follow-up video. If you cannot wait check out this website: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Also check out the following video produced as part of research by Douady and Couder about how simple displacement at the center of a plant gives rise to Fibonacci numbers of spirals https://youtu.be/U-at-y3MicE The paper itself can be found here https://www.math.ntnu.no/~jarlet/Douady96.pdf Another very interesting approach by Levitov involves a magnetic cactus, vortices in superconductors and the fabulous Farey numbers: http://www.ams.org/samplings/feature-column/fcarc-phyllotaxis#2$