Intuition of Euler's Formula WITHOUT Taylor Series

Here we discuss how to come across Euler's formula without needing to use Taylor expansion, instead exploring the dynamics of a particle undergoing circular motion. Enjoy :3

[Fernsaur]

Amazing and informative video, loved the presentation 😊

[stevenlin]

Another way to motivate Euler’s identity(this is how I teach my son) is to show the unit complex circle satisfies the law of exponent and must satisfy the form of exp(kt). Then, from physics we know if we assume an object traveling around the unit circle according to exp(kt) with unit speed whose velocity vector must be perpendicular to the position vector, then we could derive that the differentiation of exp(kx) must be k*exp(kx) and k must be the imaginary i, since the complex multiplication of i to a complex number is a 90-degree rotation. I guess this is the most intuitive approach to “see” Euler’s identity in a heuristic way, combining both the elementary physics and calculus.

[vinaynk]

The real tricky part is showing that exp(jx) is a complex number of the form a + jb.

[josecarlosribeiro]

Hello Dear Professor! Congratulatiobs for your great class and skills! Superbe! The back music teletransport tô US Enterprise directo to deep space! Thank's so much! Jacareí-Sao Paulo -Brazil.

[arshakdavidian]

If you take out of the brackets the i in the first derivative in the part of the exercise, where the right-hand side of Euler's formula is used (changing the -sinωt to +i^2*sinωt), you can easily demonstrate that in both cases the first and second derivatives of Z are ω*R and -ω^2*R.
I think that in this case, even only the first derivative should be enough to prove that both sides of the equation are indeed the same. Now, I know that if the derivatives of two functions are the same it does not necessarily mean that the higher order functions must be the same. But here we are not trying to integrate the derivative of an unknown function, but we are differentiating ourselves two functions and we know for certain that none of them have the addition of a constant involved, which could tip the balance. Am I thinking wrong?

[ty]

Not feeling really good about a velocity having an arguement of radians (units of angular revolution). Do u think the Max Plank could have left out the angular velocity moment from his constant? Think Euler was happy with conservation of units in this treatment?
Not feeling really good about an acceleration on a particle producing uniform speed much less uniform “velocity”.
No … your imprecision is convention here… dont u agree?
Not feeling really good about your treatment of real relations of force acceleration momentum and position. Convention is inconsistent so maybe its wrong huh… dont accept it simply because it is maths convention bro. Something aint right.