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Prove that 7 + 17*sqrt(17) is Irrational

Date: 09/22/2002 at 20:57:53
From: mark hailey
Subject: Proving 7+17sq.root of 17 is irrational

I need to prove 7 + 17*sq.root of 17 is irrational. I know to set it 
equal to a/b, and we also have already proven that any square root of 
a prime number is irrational.


Date: 09/22/2002 at 22:03:59
From: Doctor Paul
Subject: Re: Proving 7+17sq.root of 17 is irrational

If you are given the fact that sqrt(17) is irrational (since 17 is 
prime), then we can easily prove that 7 + 17*sqrt(17) is also 
irrational.

Proof by contradition:

Suppose that 7 + 17*sqrt(17) is rational.  Then write:

   7 + 17*sqrt(17) = p/q, 

   where p and q are integers and gcd(p,q) = 1.

Then 

   17*sqrt(17) = p/q - 7 

               = (p-7*q)/q

Then 

      sqrt(17) = (p-7*q)/(17*q), 

      which is rational.

But we already know that sqrt(17) is irrational. So we have a 
contradiction. It must then be the case that our assumption is false, 
and we conclude that 7 + 17*sqrt(17) is irrational. This was to be 
shown.

A similar method will show that a + b*sqrt(p) is always irrational 
when p is a prime and a and b are integers (or even rational, for that 
matter).

I hope this helps. Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Number Theory

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