Why Does e Appear in Nature So Often?

[quasar987]

oops there's an extra m in my topic title, I was going for why e.

I believe most the the exponential equations in physics come from the fact that a^[f(x)] = a^[f(x)] * lna * df/dx but the book I had in my first calculus class didn't had a proof for that.

Does anybody have one? And most importantly, why e? Does that number represent anything special; is it a certain ratio like pi or anything like that? It really seem to be coming out of nowhere for me. The only definitions I've seen are all unintuitive: "e is defined as the integral from there to there of this" or "e is the number such that [such and such]", etc. But why does it appear in nature so often??

(If you know a similar thread exists, tell me because I didn't find one.)

 

[Integral]

I think that there is significance in this relationship

1 = \int ^e _1 \frac {dx} x

Edit, fixed typo, now I'll add a bit of verbiage.

Think about that, the area under the inverse curve between 1 and e is equal to 1. This is a pretty fundamental definition of e.

 

[Integral]

Please do not double post.

See this thread.