• Aug 27th 2011, 07:26 PM
MathsNewbie0811
Hi,

Prove or disprove that

a. sum of two distinct irrational numbers is irrational.
b. Product of two distinct irrational numbers is irrational
• Aug 27th 2011, 07:30 PM
Prove It
Re: I need help for these questions
Quote:

Originally Posted by MathsNewbie0811
Hi,

Prove or disprove that

a. sum of two distinct irrational numbers is irrational.
b. Product of two distinct irrational numbers is irrational

a) Use a simple counterexample, like $\displaystyle \sqrt{2} + \left(-\sqrt{2}\right)$

b) Use a simple counterexample, like $\displaystyle \sqrt{2} \times \sqrt{2}$...
• Aug 28th 2011, 04:44 AM
mr fantastic
Re: I need help for these questions
Quote:

Originally Posted by Prove It
a) Use a simple counterexample, like $\displaystyle \sqrt{2} + \left(-\sqrt{2}\right)$

b) Use a simple counterexample, like $\displaystyle \sqrt{2} \times \sqrt{2}$...

I suspect the OP's use of the word 'distinct' implies 'different'.

@OP: $1 + \sqrt{2}$ and $1 - \sqrt{2}$ provide the necessary counter example for disproving the statements in both (a) and (b).
• Aug 29th 2011, 03:08 AM
HallsofIvy
Re: I need help for these questions
Or simply $\sqrt{2}*\frac{1}{\sqrt{2}}= \sqrt{2}\frac{\sqrt{2}}{2}$ which was probably what Prove It intended.
• Sep 3rd 2011, 12:18 PM
Tinyboss
Re: I need help for these questions
I've always liked this proof that an irrational to an irrational power need not be irrational: consider $\sqrt2^{\sqrt2}$. If that's rational, then we have our counterexample; otherwise, a counterexample is $\left(\sqrt2^{\sqrt2}\right)^{\sqrt2}={\sqrt{}2}^{ \sqrt{2}\sqrt{2}}=\sqrt2^2=2$.