Proving e is Irrational

Date: 11/19/97 at 21:42:47
From: Josh Decker
Subject: Proving e is irrational

For my Calculus II class I have to prove that e is irrational.  My 
professor suggested using a proof by contradiction, but I don't 
understand how to do it.  Your help would be greatly appreaciated.

Date: 11/20/97 at 15:21:17
From: Doctor Wilkinson
Subject: Re: Proving e is irrational

I'll give you a couple of hints.

You need to use the formula

   e = 1 + 1/1! + 1/2! + 1/3! +...

Suppose e is rational, i.e. e = a/b, where a and b are positive 
integers. Multiply both sides of the equation by b!.  This gives you

   b!e = b! + b!/1! +... + b!/b! + other terms

Since e = a/b by assumption, be is an integer, so b!e is an integer.  
So are the terms of the series up to b!/b!.  Now see if you can show 
that the sum of the remaining terms is less than 1.  This will give 
you a contradiction, since the difference of two integers cannot be a 
number between 0 and 1.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/