# Intuitive Understanding of the constant “ee”

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”?

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 $e$.