system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Why does Euler's Number (e) have so many unique applications?

I am a CE student who is graduating soon. I use e every day for so many different applications, but they usually boil down to modeling exponential growth and decay, modeling complex sinusoids (which seem unrelated to me but somehow e applies to both through Euler's Formula), and transforms such as the Laplace. Could someone explain how the properties of e all fit together to make these applications work? The best answer I can come up with on my own is that Euler was a dark wizard who caused nature to revolve around e.
gpt
I like your dark wizard theory, but the answer is, sadly, much simpler.

Imagine that you draw a curve on a coordinate system - never mind the equation right now, just draw the line - which has the peculiar property that, for any value of x, the y-value of the line is also equal to the slope of the line.  That would be a pretty interesting line, wouldn't it? Among other thing things, if the y-value is always equal to the slope, it would have the interesting property that f(x)=f'(x). Then you can imagine that you would want to use this function as a sort of fundamental building block to build *other* functions where some part of the function remains the same after integration or differentiation.

Are you following so far?  Do you see where this is going, or should we push it a little further?