Prove that any non zero rational number can be written as the product of two irrational numbers

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PropositionEvery non-zero rational number can be expressed as aproduct of two irrational numbers.Proof.This proposition can be reworded as follows:Ifris a non-zerorational number, thenris a product of two irrational numbers. In whatfollows, we prove this with direct proof.Supposeris a non-zero rational number. Thenr=abfor integersaandb. Also,rcan be written as a product of two numbers as follows:r=p2·rp2.We knowp2is irrational, so to complete the proof we must showr/p2is also irrational. To show this, assume for the sake of contradiction thatr/p2is rational. This meansrp2=cd

Let's see how we can modify your argument to make it perfect.

First of all, a minor picky point. You wrote $$qy=\frac{a}{b} \qquad\text{where $a$ and $b$ are integers, with $b \ne 0$}$$

So far, fine. Then come your $x$ and $z$. For completeness, you should have said "Let $x$, $z$ be integers such that $q=\frac{x}{z}$. Note that neither $x$ nor $z$ is $0$." Basically, you did not say what connection $x/z$ had with $q$, though admittedly any reasonable person would know what you meant. By the way, I probably would have chosen the letters $c$ and $d$ instead of $x$ and $z$.

Now for the non-picky point. You reached $$\frac{x}{z}y=\frac{a}{b}$$ From that you should have concluded directly that $$y=\frac{za}{xb}$$ which ends things, since $za$ and $xb$ are integers.