Question

why does constant "e" has that specific value? why isn't it bigger number or smaller or a rational number?, why does it work for growth rates?

Answer 1

e is defined like this:
\[\lim_{x \rightarrow \infty}(x+1/x)^x\]
And the result of that limit is just e, you can try it on your calculator, try to put x=1,000,000 and tell us what do you get.

Answer 2

@santistebanc Was that useful?

Answer 3

its good but is there an extended explanation, I don't quite understand it like that

Answer 4

Well, consider the binomial formula:\[(1+x)^n=\sum_{i=0}^{\infty}\left(\begin{matrix}n \\ k\end{matrix}\right)x^k\]Apply that to the definition of e given above. With a little ingenuity and a lot of knowledge about what (n k) means, you should arrive at the following definition:\[e=\sum_{i=0}^{\infty}1/i!\]This definition is much more useful, and can be used to prove things like the fact that e is irrational.
http://en.wikipedia.org/wiki/Proof_that_e_is_irrational