The golden ration known as the most irrational number. Is there proof?

Question

I do work on continuous fractures, I came across that the golden ratio is the most irrational, is there any proof of that?

Answer 1

We can prove the irrationality of the golden ratio by contradiction.

To do this, let the golden ratio, \(\phi\), be rational.

We know \(\phi>1\) so if it is rational, we could write
\[
\phi=\frac{a}{b}
\]
where \(a>b>0\) are integers and \(\operatorname{gcd}(a, b)=1\). Then using the relation \(\frac{1}{\phi}=\phi-1\) gives
\[
\frac{b}{a}=\frac{a-b}{b}
\]
which is a contradiction since \(\operatorname{gcd}(a, b)=1\) by construction and \(a>b\)