Does $e$ have a geometric representation? Just like $\pi$ is the ratio of a circle's circumference to its diameter? I know that the tangent line to the function $e^x$ has a slope of $e^x$ at that point, but is there some other geometric representation? Thanks!
 A: Consider the area under the hyperbola $y=\frac{1}{x}$, above the $x$-axis, from $1$ to $a$. This area is $1$ precisely if $a=e$. 
A: In this article http://arxiv.org/abs/0704.1282, Jonathan Sondow describes a geometric construction of the number $e$ that's different in flavor from the other answers. The idea is that his construction is a geometric representation of the identity $$\sum_{i=1}^\infty\frac{1}{n!}.$$
It's very readable for anyone who's familiar with convergence of sequences, so anyone who's taken and understood second-semester calculus.
A: As $\int_1^e \dfrac{1}{x} dx = 1$, we can say that $e$ is the length along the $x$-axis giving the area of $1$ beneath the curve $y= \dfrac{1}{x}$. (See picture.)  
A: Normally we work with this as follows: we introduce the $\ln$ function as the function that measures the area bellow the hyperbola $xy=1$ from $x=1$ to some other point $x=t$ so that we can define it as:
$$\ln t=\int_1^t \frac{dx}{x}$$
Then you prove that this function possess the properties that you want. There's a number $x$ then such that you have $\ln x = 1$ and this number is what we call $e$ so that we have:
$$\ln e=\int_1^e \frac{dx}{x}=1$$
Now you have defined this number $e$ simply as the coordinate of the point $x$ such that the area bellow the hyperbola $xy=1$ from $x = 1$ to $x=e$ is unity. With some tricks them you can find estimates to this number and even show that it's transcedental.
For more information on the subject see Spivak's Calculus.
A: I don't think it's just a matter of the tangent line at the general point having slope $e^x$.  Rather, as the graph crosses the $y$-axis, the slope is exactly one.  In fact, this gives us a way to approximate $e$ in the first place.
Examine the graphs of functions of the form $f(x)=a^x$.  As $a$ increases, these get steeper.  When $a=2$, the slope of the graph as it crosses the $y$-axis is around $0.69$.  When $a=3$, the slope of the graph as it crosses the $y$-axis is about $1.09$.  There is a very special number between $2$ and $3$ where the slope is exactly $1$.  This number is probably a bit closer to $3$ than to $2$.

