Prove that 7 + 17*sqrt(17) is IrrationalDate: 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/ |
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