Whence the number ‘e’?

is normally defined to be the base of natural logarithm. Equivalently, is defined to be the base of the natural exponential function.

What does it mean for a logarithm or an exponential function to be “natural”? Why is the base called natural? It turns out that is irrational (transcendental, actually). So, why couldn’t mathematics designate any other easy number, say 2, as the natural base, rather than the seemingly weird ? It is the case that the logarithm to base and exponential function to base are first defined, and logarithms and exponential functions to any other base are expressed in terms of these. Why is this the case?

The answer lies in calculus touching several aspects from functions to integration and differentiation. We will discuss these aspects here.