Answer to Question #135031 in Trigonometry for Kokok

Question

Write the Golden Ratio up to 30 decimals places.

Answer 1

Two quantities a and b are said to be in the golden ratio φ if

$\frac{a+b}{a}=\frac{a}{b}=\varphi$

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in$\frac{b}{a}=\frac{1}{\varphi},$

${\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}$ b/a = 1/φ,

$1+{\frac {1}{\varphi }}=\varphi$

$\varphi +1=\varphi ^{2}$

${\varphi }^{2}-\varphi -1=0.$

Using the quadratic formula, two solutions are obtained:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

${\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,749\,894\,848\,204\,586\,834\,365\dots$

and

${\frac {1-{\sqrt {5}}}{2}}=-0.618\,033\,988\,7\dots$

Because φ is the ratio between positive quantities, φ is necessarily positive:

$\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,749\,894\,848\,204\,586\,834\,365\dots$