Golden ratio

If there is a longer line segment $\scriptstyle a\$ and and a shorter line segment $\scriptstyle b\$, and if the ratio between $\scriptstyle a+b\$ and $\scriptstyle a\$ is equal to the ratio between the line segment $\scriptstyle a\$ and $\scriptstyle b\$, this ratio is called golden ratio. The value of the golden ratio is $\scriptstyle \Phi ={\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}=1{,}618033988\dots$

Properties

If $\scriptstyle {\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}$ it follows that $\scriptstyle {\frac {a}{b}}={\frac {a+b}{a}}=1+{\frac {b}{a}}$

With $\scriptstyle \Phi =1+{\frac {1}{\Phi }}$ we could derive the infinite continued fraction of the golden ratio: $\Phi =1+{\frac {1}{\Phi }}=1+{\frac {1}{1+{\frac {1}{\Phi }}}}=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{\Phi }}}}}}=\dots =1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\dots }}}}}}$