My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of $\displaystyle \frac{1}{2}n$ and write the irrationals in the form of $\displaystyle \frac{1}{\sqrt{2}}n$, where n is rational.

Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?