I am new to proof-writing and trying to make sense of what constraints are there when making assumptions when proving by contradiction.
When setting up the proof by contradiction, we assumed three things:
1) $\sqrt{2}$ is rational.
2) there exist natural numbers $p$ and $q$ such that $\sqrt{2} = \frac{p}{q}$.
3) both $p$ and $q$ have no common factors.
We then go on to show that both $p$ and $q$ are even, contradicting the assumption that $p$ and $q$ have no common factors. There is nothing in the definition of rational numbers that requires p/q to be in irreducible form. So why do we make assumption 3) in the first place?
Any clarification is much appreciated.
The only assumption is (1) $\sqrt{2}$ is rational. Since any rational number can be written as $\frac{p}{q}$, for some $p,q \in \mathbb{N}$, (2) is a consequence of (1). Without loss of generality, we can pick $p,q$ such that they don't have a common divisor. Suppose their greatest common divisor is $d \in \mathbb{N}$ with $d \neq 1$. Then, we can write $\sqrt{2} = \frac{\frac{p}{d}}{\frac{q}{d}}$ to get two natural numbers $\frac{p}{d}, \frac{q}{d}$ that satisfy (3). So, our only assumption towards the contradiction is (1), which is what we find to be false at the end of the proof.