Math Help - Proofs about irrational numbers.

Hi,

Prove or disprove that

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

2. Re: I need help for these questions

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 $\displaystyle \sqrt{2} \times \sqrt{2}$...

3. Re: I need help for these questions

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).

4. 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.

5. 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 $\left(\sqrt2^{\sqrt2}\right)^{\sqrt2}={\sqrt{}2}^{ \sqrt{2}\sqrt{2}}=\sqrt2^2=2$.