# Proving a claim using a counter example

$p,q\in\mathbb{Q}$ such that $(q\neq 0)$, then $\frac { p }{ q }$ is rational.

Steps I took and my thoughts on this:

This seems awfully obvious, but yet I can't seem to organize my thoughts in any way to construct a formal mathematical proof. The most I can think of is as follows:

let $p=\frac { a }{ b }$ and $q=\frac { m }{ n }$; such that $a,b,m,n\in \mathbb{Z}$ and $(m\neq 0$, $b\neq 0$, $n\neq 0)$

$$\frac { p }{ q } =\frac { \frac { a }{ b } }{ \frac { m }{ n } } =\frac { a }{ b } \cdot \frac { n }{ m } =\frac { an }{ bm }$$

$an$ and $bm$ are both integers because $a,b,m,n$ are all integers.

I imagine that this is way off. Please put me on the right path, and forgive my amateur attempt at a proof. I have a lot to learn about writing mathematical proofs, and I am just starting.

• This looks fine. Although, you also need the condition, $b\ne 0$. – Tim Raczkowski Oct 16 '15 at 0:09
• A counterexample can only show that a statement is false. You must be misunderstanding something. – Tim Raczkowski Oct 16 '15 at 0:12
• You are misreading the sentence. The phrase "using a counter example" is in the same clause as "disprove. – Tim Raczkowski Oct 16 '15 at 0:16
• Read as: (Prove) or (disprove using a counter example). – hardmath Oct 16 '15 at 0:16
• Yep that does it. Just one more thing: add "$\frac p q =$" to the front of the chain of equalities -- a nicety. Well done. – BrianO Oct 16 '15 at 1:10