Euler's formula and continuous rotations | Complex numbers episode 3

Video Description

#complexnumbers #euler #calculus #taylorseries How can we understand Euler's formula? This video offers no less than 3 different explanations: using derivatives and velocities, using Taylor series, or by stringing many tiny rotations after each other. Get access to the other videos in this series on Patreon: https://www.patreon.com/user?u=86649007 Some of you have pointed out that I use the phrase "straight angle" incorrectly. In English, an angle of 90 degrees is called a "right angle". I apologize for the confusion. Whenever I say "straight angle" in this video, please try to hear "right angle" instead ;-) Some interesting links that were referenced in the video: [JHAN 1] https://www.youtube.com/watch?v=TLgZit1HTxA Explains Euler using the mechanics of circular motion. [3B1B 1] https://www.youtube.com/watch?v=v0YEaeIClKY Another short but good explanation based on circular motion. [3B1B 2] https://www.youtube.com/watch?v=mvmuCPvRoWQ Explains Euler's formula using group theory. The exponential function converts between addition and multiplication. [WOO 1] https://www.youtube.com/watch?v=8Y2eIPCg7uc In this 8 part lecture series, Eddy Woo shocks his students with Euler's formula. [WOO 2] https://www.youtube.com/watch?v=QGso1gxgGBQ [WOO 3] https://www.youtube.com/watch?v=TejELKLWNA8 Another lecture by Eddy Woo, where he derives Euler's formula. [M 1] https://www.youtube.com/watch?v=-dhHrg-KbJ0 The Mathologer video that explains continuous interest on a bank account. [KON 1] https://www.youtube.com/watch?v=-4G8JLIOHMI Sander Konijnenberg: A more detailed explanation of Euler, but not as visual/geometric as the others. Mostly just formulas. 0:00 Introduction 1:19 Powers of complex numbers 5:21 Euler's formula 7:19 Exponentials of complex numbers 8:30 Velocity on a circle 11:15 Taylor series 12:05 Many tiny rotations 17:08 Lie groups & Lie algebras 18:47 Complex roots of unity This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )