Show that the product of a non-zero rational number an an irrational number is irrational

First I tried direct proof;

Let assume $r \in Q$ and k $\in [R-Q]$ and show that r*k $\in [R - Q]$, so $r.k=\frac{ak}{b}$, where a,b $\in Q$, but as you see it creates the same problem, meaning $a*k$, so I stuck.

I tried to proof by contrapositive and contradiction, but I still didn't find the way.

Note: I just know these 3 proving methods -direct, contrapositive and contradiction - for now; therefore if you used just these, I would appreciate that.


If $rk=q$, and $q$ is rational, then $k=\frac{q}{r}$, which is rational.

Does that help? Do you see which proof method this uses?

  • $\begingroup$ it does absolutely, thanks . $\endgroup$ – onurcanbektas Oct 24 '16 at 11:35

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