Proving e is IrrationalDate: 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/ |
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