Euler's formula with introductory group theory

Ah, a vaguely wondered if it had something involving Cauchy sequences, I think that was that limit! > The way we extend exponentiation to complex exponents is IMHO much less straightforward. I think it depends on how much you're used to dealing with complex numbers. In college, I was always taught to prove that Euler formula by replacing ix for x in that series, and then noting that the alternating signs and presence/absence of i in the terms allowed you to separate it into two series for cosine and sine. That always felt awkward, like there was no way anyone could just come up with that naturally. Many years later I found that construction with graphing e^(ix) by taking small steps using f(x+dx)=f(x)+f'(x)*dx, and everything clicked: how exponentials work in the real axis is pretty different from the imaginary axis, but both are completely intuitive and unavoidable once you understand that. Even later I "discovered" the connection with group theory[1]; that one still blows my mind. [1] This is a really nice explanation: ▶ this video

The repeated "many folding", which would be better visualized as tendition, exponentiation pattern also breaks with fractions & at the most basic negative numbers: ▶ this video

IMHO, the "deeper truth" is addition is translation and multiplication is rotation/stretching (at least when it comes to rings and fields). It involves getting a geometric understanding of e to the pi i, and 3blue1brown explains it better than I could: link [6.28 min] link [3.14 min] ▶ this video [25 min]

I am unable to answer your questions for you, but I wanted to talk on the topic because it is something I've been thinking a lot about lately There recently was a lot of talk and reporting surrounding YouTube 'changing their algorithm' and hurting a lot of YouTubers I watch frequently One of my thoughts was that unfortunately the best solution would probably mean content creators would need to seek their own advertising'sponsorships I can tell you the only ads I have clicked on in any recent memory are all, every single ad, of the ones that 3blue1brown places at the end of thaer videos(o) Considerate effort like this really made me think how inconsiderate and detrimental the current hands off one size fits all ad networks are Some content creators will surely protest 'im a creator stead a salesman' which is a fair complaint but I'd argue 3blue1brown really brings an artistry to the personal ad network the channel is creating.. self curated ads has the potential to be another medium of self expression (o) ▶ this video .. an aside: the ad that YouTube ran in front of the video I linked when I first clicked on it was offensive and jarring, it depicted an animated Albert Einstein literally scribbling on a black board while a student looked on in dread and three words came in the screen, two of which I can remember 'math ___ sucks' the middle was either: teachers, lessons, or tutors; this ad actually upset me.. contrast that with this great 3blue1brown ad(i) and I am more confident in

That whole channel is amazing! I can start just listing all the videos by virtue of picking out awesome ones, but check out the one he did on Euler's formula[1]. Blew my mind. [1] ▶ this video