Definition of irrational numbers

1. Let's suppose that $\sqrt{7}=\dfrac{p}{q}$ where $p$ and $q$ are integers without factors in common. We multiply by $q$ and raise the expression to the square, obtaining; $$7q^2=p^2$$

If we do the factorization in prime numbers we see that on the left side there is a odd number of sevens and on the right side an even number. As such, we can say that a rational expression of $\sqrt{7} does not exist.$

2. If $3\pi$ was rational we would have $3\pi=\dfrac{p}{q}$, where $p$ and $q$ are integers. Then we would have $\pi=\dfrac{p}{3q}$ and $\pi$ would be rational, which is clearly false.

So, we can say that $3\pi$ is not rational.