TITLE: Why aren't all square roots irrational?
QUESTION [18 upvotes]: The more known proof of square root of $2$ is by contradiction when we assume it can be expressed as an irreducible fraction and later finding that it isn't irreducible, but... if we assume the same conditions for example: square root of $9$ we find that it isn't irreducible either. So, what's the trouble here?

REPLY [2 votes]: See this proof that 
if $n$ is not a perfect square
then $\sqrt{n}$
is irrational:
Follow-up Question: Proof of Irrationality of $\sqrt{3}$
The proof starts by saying that
if $n$ is not a perfect square
then there is a $k$ such that
$k^2 < n < (k+1)^2$.
The proof breaks down if
$k^2 = n$.