That one's "just" a special case of how complex numbers happen to work. I think the really cool relationship between e and pi is the fact that the Gaussian integral acts as a fixpoint/attractor when sampling and summing data from any distribution (this is the central limit theorem):
∫(−∞ to ∞) e^(-x²) dx = √π
I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.
∫(−∞ to ∞) e^(-x²) dx = √π
I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.