Famous Theorems of Mathematics/e is irrational
The series representation of Euler's number e
can be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function ey evaluated at y = 1.
Summary of the proof
This is a proof by contradiction. Initially e is assumed to be a rational number of the form a/b. We then analyze a blown-up difference x of the series representing e and its strictly smaller bth partial sum, which approximates the limiting value e. By choosing the magnifying factor to be b!, the fraction a/b and the bth partial sum are turned into integers, hence x must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1. From this contradiction we deduce that e is irrational.
Suppose that e is a rational number. Then there exist positive integers a and b such that e = a/b.
Define the number
To see that x is an integer, substitute e = a/b into this definition to obtain
The first term is an integer, and every fraction in the sum is an integer since n≤b for each term. Therefore x is an integer.
We now prove that 0 < x < 1. First, insert the above series representation of e into the definition of x to obtain
For all terms with n ≥ b + 1 we have the upper estimate
which is even strict for every n ≥ b + 2. Changing the index of summation to k = n – b and using the formula for the infinite geometric series, we obtain
Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so e must be irrational.