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