Intuitive Understanding of the constant "$e$"
I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you explain e concisely?
It's a rather beautiful number, but when friends have asked me "what is e?" I'm usually at a loss for words -- I always figured the math explains it, but I would really like to know how others conceptualize it, especially in common-language (say, "English").
related but not the same: Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"?
Solution 1:
If someone asks me, "what is $e$?" I sketch the graph of $y=1/x$, draw a line segment from $(1,1)$ on the curve down to $(1,0)$ on the $x$-axis, and ask, how far to the right do I have to draw another vertical segment to rope off an area of 1? Anyone who is familiar with the idea of graphing a function can appreciate that definition, and it's not surprising that something with such a down-to-earth definition is going to turn up in lots of other places in Mathematics. And anyone who knows Calculus can be shown that all the other properties of $e$ and of $e^x$ and of $\log x$ can be derived from this one property of $e$.
The yellow area equals the red one:
Solution 2:
For the somewhat-calculus-literate, your "related but not the same" question is what I'd go for: $e$ is the number for which the exponential function with that base is its own derivative.
Without calculus, I'd go for the notion of compound interest: With a rate $r$ per period, compounded $n$ times per period, $A$ grows to $A(1+\frac{r}{n})^n$ after 1 period; as $n\to\infty$, $A(1+\frac{r}{n})^n\to Ae^r$.