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[Community Question] Calculus: Why is Euler's number 2.718 and not any thing else?

One of our user asked:

Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$ ???

I know that e is the base of natural logarithms, I know about areas on hyperbola xy=1 and I know it's formula : $$e =\sum_{n=0}^\infty \frac{1}{n!}$$
And I also know it has many other characterizations.
But, why is e equal to that formula (which sum is approximately $\mathtt 2.71828$) ???
I googled that many times and every time it ends in having "e is the base of natural logarithms",
I don't want to work out any equations using e without understanding it perfectly.

Thanks in advance.


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