I believe the contrapositive method should be correct but i get,
The contrapositive of this statement should be,
(If $\sqrt{x}$ is rational, then $x$ is rational)
Then I end up with $x=\frac{m^2}{n^2}$ for $m,n$ to be integers and $n$ does not equal 0.
Is it careless to be done with the proof since we are saying that $x=\frac{m^2}{n^2}$ is a rational number therefore the contrapositive is true, therefore the original statement is also true?
Best Answer
$[p\implies q]\iff [\neg q\implies\neg p]$
If you prove the contrapositive of a statement, that completes the proof of the statement, because a statement is true if and only if its contrapositive is true (draw a truth table and verify this).
The Proof by Contradiction also can be used:
Let $x$ is irrational and assume $\sqrt x$ is rational. Then you can arrive at a contradiction to $x$ is irrational.