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

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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?

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 $a>b$
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