#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]    • Why do trig functions appear in Euler...  
Explains Euler using the mechanics of circular motion.

[3B1B 1]    • e^(iπ) in 3.14 minutes, using dynamic...  
Another short but good explanation based on circular motion.

[3B1B 2]    • Euler's formula with introductory gro...  
Explains Euler's formula using group theory. The exponential function converts between addition and multiplication.

[WOO 1]    • The Most Beautiful Identity (1 of 8: ...  
In this 8 part lecture series, Eddy Woo shocks his students with Euler's formula.

[WOO 2]    • Informal Proof of Euler's Formula (1 ...  
[WOO 3]    • Informal Proof of Euler's Formula (2 ...  
Another lecture by Eddy Woo, where he derives Euler's formula.

[M 1]    • e to the pi i for dummies  
The Mathologer video that explains continuous interest on a bank account.

[KON 1]    • (SoME1) Imaginary numbers with real a...  
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/... )