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Proof That the Cube Root of 3 is Irrational

Date: 05/22/2000 at 23:16:44
From: Sajesh Singh
Subject: Proof of an irrational number


I need help with the following problem:

Show that the cube root of 3 is irrational.

Thank you for your help
Sajesh Singh

Date: 05/23/2000 at 02:39:58
From: Doctor Floor
Subject: Re: Proof of an irrational number

Hi, Sajesh,

Thanks for writing.

I will prove that the cube root of an integer - such as 3 - is only 
rational if it is an integer. I will use an indirect proof.

To do this, suppose that the cube root of an integer N is rational, 
but not an integer. We can write this cube root as a/b in such a way 
that GCD(a,b) = 1 (a and b are coprime) and b > 1. Since GCD(a,b) = 1, 
we can't simplify the fraction a/b.

We derive N = a^3/b^3. But since a and b are coprime, a^3 and b^3 are 
coprime, too, and thus the fraction a^3/b^3 can't be simplified. And 
since b > 1 so too b^3 > 1. But that means that a^3/b^3 is not an 
integer, while N is an integer. And we have a contradiction.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum   
Associated Topics:
College Number Theory
High School Number Theory

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