Proof of Irrationality of e

The base of the natural logarithm,occurs naturally in maths. For example the solution to the differential equationis- the equation of exponential growth - whereis the value ofatlikeand is irrational so cannot be written as the quotient of two natural numbers. We can prove this quite easily using the Taylor series ofabout

In this series expansion put

Suppose thatthen

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 termand common ratioso has sumcontradicting that

is an integer so e is not rational.