Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Hi,

I need help with the following problem:

Show that the cube root of 3 is irrational.

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
http://mathforum.org/dr.math/

Associated Topics:
College Number Theory
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search