TITLE: Two questions about Euler's number $e$
QUESTION [7 upvotes]: I am on derivatives at the moment and I just bumped into this number $e$, "Euler's number" . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of any point is 1. Also it is an irrational ($2.71828\ldots$) number that never ends, like $\pi$. 
So I have two questions, I can't understand 

What is so special about this fact that it's slope is always 1? 
Where do we humans use this number that is so useful, how did Mr Euler come up with this number? 

and how come this number is a constant? where can we find this number in nature?

REPLY [4 votes]: Just a slight correction, as Jon Claus notes about the derivative of $e^x$:
what you may be remembering is that "$e$ is the unique real number such that the value of the derivative (slope of the tangent line) of the function $f(x) = e^x$ at the point $x = 0$ is equal to $1$.
See the Wikipedia article on Euler's number $e$ for more fascinating information:

The number e is the unique positive real number such that
$$\frac{d}{dt}e^t = e^t.$$
The number e is the unique positive real number such that
$$\frac{d}{dt} \log_e t = \frac{1}{t}.$$
The following three characterizations can be proven equivalent:
The number e is the limit
$$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$
Similarly:
$$e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}$$
The number e is the sum of the infinite series
$$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
where $n!$ is the factorial of n.
The number e is the unique positive real number such that
$$\int_1^e \frac{1}{t} \, dt = 1.$$



The number $e$ is of eminent importance in mathematics, alongside $0, 1, \pi, \;\text{and}\; i.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity: $$e^{i\pi} + 1 = 0$$ Like the constant $π, e$ is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients.