Proof Phi is Irrational by using another Irrational Number
It is known to mathematicians that Phi (the golden ratio) is irrational number.
The value of Phi is $\frac{(1+\sqrt5)}2$. The task is to use another irrational number (not $\sqrt5$) to proof the irrationality of Phi.
You don't need to use any irrational numbers. $\phi$ is a root of the polynomial $z^2 - z - 1$. Using the Rational Root Theorem, it's easy to show that this has no rational roots.
If you really want to, you could change this into a proof that "uses" another irrational number such as $1/\phi$.