The base of the natural logarithm,
occurs naturally in maths. For example the solution to the differential equation
is
- the equation of exponential growth - where
is the value of
at
like
and 
is irrational so cannot be written as the quotient of two natural numbers. We can prove this quite easily using the Taylor series of
about
In this series expansion put
Suppose that
then
Split the summation into two parts.
The first term on the right is an integer and so must the second term be but
is a geometric series with first term
and common ratioso has sum
contradicting that
is an integer so e is not rational.