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You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers, then $p + (q-p)/\sqrt 2$$$p + \frac {q-p}{\sqrt 2}$$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers, then $p + (q-p)/\sqrt 2$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers, then $$p + \frac {q-p}{\sqrt 2}$$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers with $p < q$, then $p + (q-p)/\sqrt 2$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are rational numbers with $p < q$, then $p + (q-p)/\sqrt 2$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers, then $p + (q-p)/\sqrt 2$ is an irrational number between $p$ and $q$.
You are making this more complicated than it needs to be. If $p,q$ are rational numbers with $p < q$, then $p + (q-p)/\sqrt 2$ is an irrational number between $p$ and $q$.
