If r is rational(r does not equal 0) and x is irrational, prove that r + x and rx irrational.

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- Jun 11th 2006, 03:39 PMNichelle14Where do I start
If r is rational(r does not equal 0) and x is irrational, prove that r + x and rx irrational.

- Jun 11th 2006, 04:14 PMThePerfectHackerQuote:

Originally Posted by**Nichelle14**

$\displaystyle r+x$ is rational.

Thus,

$\displaystyle r+x=\frac{p}{q}$

Then,

$\displaystyle x=\frac{p}{q}-r$

But, $\displaystyle r$ is rational,

Thus,

$\displaystyle x$ must be rational.

A contradiction.

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Assume that,

$\displaystyle rx$ is rational.

Thus,

$\displaystyle rx=\frac{p}{q}$

Now, $\displaystyle r\not = 0$ thus,

$\displaystyle x=\frac{p}{rq}$

Thus, $\displaystyle x$ is rational.

A contradiction. - Jun 11th 2006, 04:40 PMNichelle14
Thank you. That makes sense. I thought of doing something with p/q, but didn't know how to proceed.

Again thank you. - Jun 11th 2006, 05:10 PMThePerfectHackerQuote:

Originally Posted by**Nichelle14**

Let us keep all you question in "Number Theory" section. I like to leave misselanous topic in that section.