Adding Rational and Irrational Numbers

Date: 11/09/1999 at 11:14:45
From: Jessica Troise
Subject: Abstract Math I

I was wondering how you could prove that a rational number added to an 
irrational number results in an irrational number.

Date: 11/09/1999 at 12:54:15
From: Doctor Wilkinson
Subject: Re: Abstract Math I

When you want to prove something is irrational you should almost 
always use an indirect proof. So suppose that x is rational and y is 
irrational. You want to show that x + y is irrational, so you assume 
it's rational. So you have

     x          rational
     y          irrational
     x + y = z  rational

What happens if you solve

     x + y = z

for y in terms of x and z?

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/