What is Euler's formula actually saying? | Ep. 4 Lockdown live math
Why Hacker News recommends it
HN recommends it as the explanation that makes complex exponentials click, especially for understanding the Fourier transform.
What it is
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.
Recommended by 1 HN commenters across 1 different months, 2020 to 2023.
What Hacker News said
Strong rec rectang · 2023-01 · on “The Fourier Transform, explained in one sentence”
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]
Recommends rectang · 2023-01 · on “The Fourier Transform, explained in one sentence”
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]
Cites beforeolives · 2021-06 · on “Is Euler’s Identity Beautiful? and If So, How?”
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]
Critical rathel · 2020-10 · on “A Programmer’s Intuition for Matrix Multiplication”
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 expon