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: $\Phi =1+{\frac {1}{\Phi }}=1+{\frac {1}{1+{\frac {1}{\Phi }}}}=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{\Phi }}}}}}=\dots$

Thus

$\Phi \ =\ 1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\dots }}}}}}$

• $\Phi \ =\ \lim _{n\to \infty }{\frac {F(n+1)}{F(n)}}$,

where $\ F_{n}$ is the n-th term of the Fibonacci sequence.

• The golden ratio is irrational and, in a sense, the hardest among irrational numbers to approximate by rational numbers. Only rational numbers are harder to approximate by other rational numbers. Thus one may say that of all irrational numbers the golden ratio is the least irrational.