TITLE: $\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$?
QUESTION [12 upvotes]: If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed the alternative characterizations on wikipedia and they all involve limiting processes in one form or another. So I was wondering: Is there a visual, easy-to-grasp way to present $e$? (Even if the presentation wouldn't be mathematically equivalent to the usual ones.)

REPLY [6 votes]: The notion of period, which is introduced by Kontsevich and Zagier, would partially give a negative answer to your question. According to this article, it is now known whether $e$ is a period or not, though it is conjecturedly not a peroid. In particular, $e$ seems not to arise as an area or a length of a geometric figure defined by an algebraic equation.
This may imply that any description involving $e$ would eventually require some extent of advanced calculus.
We may circumvent this problem by arguing that the concept of $e$ arises rather naturally by considering some real-world problems. For example, we can think of compound interest:
Given an annual interest $100r$% with compounding period $1/n$ year, the relative value at the end of a year is
$$ \left( 1 + \frac{r}{n} \right)^{n}, $$
which converges to $e^{r}$ as $n \to \infty$ (that is, the period of interest goes to zero).
Here is another, though odd, characterization of $e$:
Suppose that a drunk man starts at $x = 0$ at time $t = 0$, and then at each second he walks to the right with a step size randomly and evenly between $0$ and $1$. That is, if $X_i$ are i.i.d. random variables having a uniform distribution on $[0, 1]$, then the position of the drunk man at time $t = n$ is $x = X_1 + \cdots + X_n$. Let $T$ be the time when his position first exceeds the point $x = 1$. Then the mean of $T$ is exactly $e$.