What makes e so fundamental?

Question

I'm a mathematician, so I am obviously blown out of the water by the beauty of Euler's Identity (e^(i*pi)+1=0).

When explaining this incredible fact to people, it's pretty easy to get them to buy that 0,1,i, and pi are fundamental numbers to mathematics. But e is more subtle. Yes, I understand its value, and many of its amazing properties, but they're a bit harder to convey (calculus feels less fundamental than algebra or geometry).

My question is: If you could sum up the beauty and fundamental importance of e in one or two sentences, what would they be?

Answer 1

The success of e came with calculus: e^x is the only function that is its own derivative AND its own integral.

d(e^x) / dx = e^x

Integral of e^x dx = e^x (+ a constant)

Answer 2

Most fundamental Christains don't believe in the rapture. Despite what some media may mistakenly claim or label, fundamentalists are some of the biggest opponents to the rapture belief. What we label as "fundies" truly isn't fundamental at all.

Answer 3

If Euler's Identity doesn't do it, nothing will.

The elegance and impact of that simple expression, relating transcendental numbers to real and imaginary numbers, is as pretty and awe-inspiring as math gets.

This is a bit arrogant, but true.

You need Russian to really "get" Pushkin.
You need math to "get" Euler.

Answer 4

As a mathematician, why would you want to limit yourself to one or two sentences? When compounding the interest of your audience, in e & ln, watch their expressions for signs of growth or decay in attention. Arch your back, when you get tired; but, don't leave your audience suspended. Expand the discussion, the beauty will be revealed by your unique expression of "What e means to me".