Why is the number e so important in mathematics?

Question

I've heard a lot about this number e. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used?

Answer 1

I answered this question recently; I will copy my previous answer for you.

Perhaps the pre-eminent reason why e is so important is that the exponential function e^x is its own derivative,

i.e. if y = e^x, then dy/dx = e^x.

e = ∑(1/n!) for n = 0 to ∞.

It is also the limit [1+(1/x)]^x as x approaches 0

And, of course, it ties together the most important mathematical constants: e^iπ + 1 = 0, which a a specific form of Euler's formula: e^iθ = cos θ + i sin θ.

This is just a short answer to whet your appetite. Go do some reading and studying, starting perhaps with M. Spivak's "Calculus">