What is Euler's formula actually saying? | Ep. 4 Lockdown live math on HackerTube: recommended by 1 Hacker News commenter across 1 month, 2020–2023.
lecture · intermediate · evergreen
A 3Blue1Brown live lockdown math episode asking what Euler's formula e^(pi i) is actually saying. It builds intuition through exponentials, rotation and vectors landing on the unit circle.
HN recommends it as the explanation that makes complex exponentials click, especially for understanding the Fourier transform.
“I was finally able to understand the role of complex numbers in the Fourier Transform.”
4 Hacker News comments link this video. Each one links back to its thread.
The key insight for me on this topic also came from 3Blue1Brown, but in a different video: it’s that e^x is NOT best thought of as repeated multiplication, but instead as the function exp(x) = 1 + x + x^2/2 + X^3/6 + x^4/24 + … After being relieved of the burden of that misconception, I was finally able to understand the role of complex numbers in the Fourier Transform. ▶ this video
in “The Fourier Transform, explained in one sentence” · View on HN
People told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation! Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did. ▶ this video
in “The Fourier Transform, explained in one sentence” · View on HN
I think that e^(i*pi) + 1 = 0 is massively overrated. Because the beauty that people find is largely based on the symbols that are present and because it doesn't really say anything meaningful beyond itself. It's not even a formula or an equation - it's a result. It's like being impressed by cos(pi) = -1. The more general formula e^(ix) = cosx + i*sinx is a lot more impressive. Both because of it's implications for the rest of mathematics and aesthetically. If you look at the way the vectors at 38:00-39:20[1] cancel out to always land on the unit circle - I can get behind calling that beautiful. [1] ▶ this video
in “Is Euler’s Identity Beautiful? and If So, How?” · View on HN
True, but I don't agree with what he said here @ 8:00 ▶ this video Is this really the case? I checked Wikipedia and MathWorld and nobody makes a distinction between e^x and exp(x). Even if it's a "white lie" for didactic reasons, I don't buy it, it will be much more confusing down the line for students. Math is about finding structures governed by some rules and then generalizing them. To me the e^x defined as repeated multiplication conceptually is the same thing as the exponential function, just over broader domain. Why? You can interpolate between integer-valued X's using geometric mean. e^3 = sqrt(e^2 * e^4). What stops you from interpolating this recursively to achieve in the limit the exponential function over real numbers?
in “A Programmer’s Intuition for Matrix Multiplication” · View on HN