I enjoyed this read. Here’s some related thoughtwork I had last week, which looks at it via Calculus.

My son asked me:

“what’s the derivative of e^x”?

I replied “it is e^x?”, inquisitively as it had been a while. Once he confirmed, I started envisioning circles and vectors moving and said “oh, that is another reason why Euler’s formula is true”.

I’m just a nerd, not a mathematician, so I didn’t completely grok my flash of insight. But simple enough to ask an LLM, and Claude built the system of differential equations showing it:

https://claude.ai/share/33033109-a0fa-4c7b-822d-ca897c442cf2