Simon was showing Dad a graph of how technology is developing exponentially, y = a^x. Dad asked for a specific value of a, and Simon said: “All exponentials are stretched out or squished versions of the same thing.” He then quickly came up with the proof (“a few lines of relatively simple algebra”). “If all exponentials are pretty much the same, that means that all exponentials have proportionately the same derivative.”
e^ipi = – 1
The idea comes from a video by Mathologer. Simon sketches a geometric definition of the Euler’s number (e) using integrals. He messed up a little with the integral notation, but corrected it later (after we stopped filming). Please see the photos below:
In this video, Simon talks about the discovery of the e constant (Euler’s Number) and how it relates to calculating annual interest rates in banking. Inspired by a Numberphile video.