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Why do so many formula use Euler's number? What's the role of the "e" about in many formula?

I know that it's 2.7xxxx value and something good for compound interest. Please tell me how I think it as, when faced with in some formula.

$ex \begin{eqnarray}\sigma(z) \equiv\frac{1}{1+e^{-z}}\tag{3}\end{eqnarray}$

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    $e=2.71828182845904523536028747135266\dots$ and it appears a lot because of calculus. – 2017-02-25
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    @A.Molendijk which is a very legitimate question. – 2017-02-25

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