system
Given the following post, respond as if you're commenting on Reddit:
human
Why is the value of e, e?

I am talking about the natural constant. I am wondering why e shows up everywhere. Why is it that very different things resemble e? For example, continuous interest has e in its equation while the amount of electrons in a capacitor has a natural logarithm (log base e). Why is this so? I assume e is just a number just like pi is pi because it is.
gpt
Quick and fairly easy answer: it comes up everywhere because it has the unique property that d e^x /dx = e^x.  The numerical value is an approximation to the limit that defines e, e^x = (1+x/n)^n, as n -> inf.  You can derive that definition if you solve the differential equation dy/dx = y by assuming y can be written as a power series Sum_i (c_i x^i), computing the coefficients recursively and showing that the series converges.

Expanding on the above, it comes up everywhere because it's the solution to a whole range of differential equations.  Simple examples: e^ax solves y' = ay, y'' = a^2 y, y''' = a^3 y etc.  More complicated cases are, let's say, riffs on this basic idea--that the derivatives of y depends somehow on y itself.