[rectang]

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.

https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=439s

[fortran77]

Or even more simply, if you know that e^x on the complex plane rotates you around the origin.

[rectang]

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.

https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s

[orangecat]

The best explanation I've seen, which 3blue1brown probably avoided because it requires calculus, is that exp is the unique function whose value at 0 is 1 and whose derivative is itself. Then by the chain rule the derivative of exp(ix) is i*exp(ix), meaning that the direction of motion is perpendicular to the current position vector. Which naturally leads to circular motion because you're always staying the same distance from the origin.

With that definition it's easy to derive the Taylor series expansion (every derivative at 0 is 1), and you can think of Euler's formula not as telling you how to evaluate exp(ix) (it's already perfectly well defined), but as an introduction of cos and sin as shorthand for its real and imaginary parts.

[nuancebydefault]

I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.

[rectang]

Indeed. The title of that 3Blue1Brown video is, “What is Euler’s Formula actually saying?”

That “easy to remember” formula had been presented to me countless times. It only made sense once I stopped thinking about it in terms of repeated multiplication.

[l-]

The repeated "many folding", which would be better visualized as tendition, exponentiation ("tendaddition") pattern also breaks with fractions & at the most basic negative numbers: https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=922s Also: https://news.ycombinator.com/item?id=28524792 "log base" would be better named untendaddition similarly division - does not necessary "separate" - untendition & unaddition - does not "draw under". Etymologos to use "given" symbols over the datum names.

[l-]

Further, the pattern matched "grouping" definitions (https://news.ycombinator.com/item?id=25278021) names have a better correspondence to the Greek scheme: diation, triation, tetration... while the "Group theory" definitions scheme lending to various geometries (hypercomplex: complex::hyperbolic, split-complex::elliptic, dual::parabolic) would match some other naming scheme.