if x in an rational number but not equal to zero and y is an irrational number. Proof that x/y is an irrational number?

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- Feb 28th 2008, 07:05 AMsthatoProof the following using denseness in reals
if x in an rational number but not equal to zero and y is an irrational number. Proof that x/y is an irrational number?

- Feb 28th 2008, 08:24 AMJaneBennet
If $\displaystyle \frac{x}{y}$ is rational, then $\displaystyle y=x\left(\frac{x}{y}\right)^{-1}$ is rational too. ($\displaystyle \left(\frac{x}{y}\right)^{-1}$ exists as $\displaystyle x\ne0$.) Contrapositively, if $\displaystyle y$ is irrational, $\displaystyle \frac{x}{y}$ must be irrational.

- Feb 28th 2008, 09:44 AMsthatoThanks
Thanks for your help Jane.(Yes)