Euler's formula with introductory group theory
Why Hacker News recommends it
Recommended as a mind-opening, visual explanation of why Euler's formula is true.
What it is
3Blue1Brown builds intuition for Euler's formula, e^(pi i) = -1, using the core ideas of group theory. It treats exponentiation geometrically as rotation and scaling instead of repeated multiplication.
Recommended by 3 HN commenters across 3 different months, 2017 to 2024.
What Hacker News said
Strong rec BucketSort · 2017-04 · on “Miniatures: Mathematical and Algorithmic Applications of Linear Algebra [pdf]”
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]
Recommends kortex · 2020-12 · on “Alternative notation for exponents, logs and roots? (2011)”
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]
Recommends moefh · 2024-07 · on “What makes e natural? (2004)”
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(
Cites justifier · 2017-05 · on “Ask HN: Anyone monetizing their sideproject with direct ads / sponsorships?”
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 of
Cites l- · 2023-01 · on “The Fourier Transform, explained in one sentence”
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]