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I do work on continuous fractures, I came across that the golden ratio is the most irrational, is there any proof of that?
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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 $a>b$
by Gold Status (31,693 points)

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