Proof Phi is Irrational by using another Irrational Number

Question

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.

Answer 1

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 $1/\phi$.

Answer 2

First, you can prove that there is no rational roots for $x^2+x-1$ using the rational root theorem. Since $\frac{1}{\varphi}$ and $-\varphi$ are roots to the equation, they are irrational, meaning that $\varphi$ itself is irrational.