Intuitive Understanding of the constant "$e$" Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.


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"?

It's easy to show that $\dfrac{d}{dx} 2^x = (2^x\cdot\text{constant})$.  And $\left.\dfrac{d}{dx} 2^x\right|_{x=0} = \text{that constant}$.
Since the graph of $y=2^x$ gets steeper as $x$ grows, the slope at $x=0$ must be less than the slope of the secant line involving $x=0$ and $x=1$.  That latter slope is 1.  Therefore the "constant" is less than 1.
By thinking about $y=4^x$ and considering the secant line involving $x=-1/2$ and $x=0$, one sees that that "constant" is more than 1.
Therefore 2 is too small, and 4 is too big, to be $e$.
For $y=e^x$, the "constant" is exactly 1.
(One can show that 3 is too big via the secant line at $x=-1/6$ and $x=0$, but the arithmetic is a bit messy.)  Similarly $2.5$ is too small, via $x=0$ and $x={}$ . . . . I don't remember which number I used here.  A positive number, obviously, and less than 1.  Messy arithmetic again.
