The number e explained in depth for (smart) dummies @Mathologer

Mathologer | The number e explained in depth for (smart) dummies @Mathologer | Uploaded 7 years ago | Updated 1 hour ago
In this follow-up video to his "e to the i pi for dummies" video the Mathologer sets out to properly explain the coolest features of the famous number e and the exponential function e^x. Find out WHY e is irrational, how you go about calculating the first 1,000,000 digits of e, WHY the exponential function e^x is its own derivative, etc.

Here are links to the videos that I refer to in this video:

e to the pi i for dummies: youtu.be/-dhHrg-KbJ0 (this is the video I summarise at the beginning)

Indeterminate: the hidden power of 0 divided by 0: youtu.be/oc0M1o8tuPo (about derivatives, among other things)

Math in the Simpsons: e to the i pi: youtu.be/Yi3bT-82O5s (this is the video that I refer to at the very end)

This week's t-shirt I made myself. Check out this wiki page about this pretty identity en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π

Thank you very much to my friend Marty Ross for proofwatching drafts of this video and helping me to get the words "just right" and to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy!

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The Korean kings magic square: a brilliant algorithm in a k-drama (plus geomagic squares)

Why dont they teach simple visual logarithms (and hyperbolic trig)?

The 15 puzzle - solving the unsolvable 19th century Rubiks square

The Kakeya needle problem (the squeegee approach)

$Why did we forget this simple visual solution? (Lills method) Todays video is about Lills method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horners form, synthetic division and a newly discovered incarnation of Pascals famous triangle. 00:00 Intro 04:14 Lills method 07:31 Free meal 09:51 Square turtles 11:39 Origami turtles 14:16 Iterative turtles 17:32 QED 24:00 Pascals turtle animation Here is the page with an implementation of Lills method for cubic polynomials that I show in the video. http://www.qedcat.com/misc/lill_method/ Its an adaptation of this webpage http://heim.ifi.uio.no/magho/lill/ (I have not been able to find out who put this together originally). The article that inspired this video is this: Thomas C. Hull, Solving Cubics With Creases: The Work of Beloch and Lill, The American Mathematical Monthly , Vol. 118, No. 4 (April 2011), pp. 307-315. Here is a link to this article on Thomas Hulls webpage: http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf Lills original paper: http://www.numdam.org/article/NAM_1867_2_6 359_0.pdf Other good references include: Polynomials as polygons by Serge Tabachnikov https://www.math.psu.edu/tabachni/prints/Polynomials.pdf Dan Kalmans book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lills method) https://books.google.com.au/books?id=JPq0pS3wrx4C&pg=PA7&source=gbs_toc_r&cad=3#v=onepage&q&f=false Thank you very much to Marty, Karl and Danil for their help with this video. One version of todays math t-shirt (Zombie addition): https://www.redbubble.com/people/manikx/works/8929883-zombie-math?p=t-shirt The piece of music at the end is called Fresh fallen snow by Chris Haugen from the free YouTube music library. Really neat 1-line Mathematica code for the generation of the Pascal turtle which appeared on Reddit after the video was posted there: Graphics[Table[Line[ReIm[Accumulate[Table[2^(-n/2)Binomial[n,k]Exp[I(4+2k-n)Pi/4],{k,-1,n}]]]],{n,0,7}]] and another nice implementation in Python (with a real turtle graphics turtle) by Alex Hall https://repl.it/repls/DeepskyblueFractalPoint Enjoy :) Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)$

Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

Euler’s Pi Prime Product and Riemann’s Zeta Function