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?
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.