Answer to Question #135031 in Trigonometry for Kokok

Question #135031
Write the Golden Ratio up to 30 decimals places.
1
Expert's answer
2020-09-28T17:12:58-0400

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

a+ba=ab=φ\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 inba=1φ,\frac{b}{a}=\frac{1}{\varphi},

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


1+1φ=φ1+{\frac {1}{\varphi }}=\varphi

φ+1=φ2\varphi +1=\varphi ^{2}

φ2φ1=0.{\varphi }^{2}-\varphi -1=0.

Using the quadratic formula, two solutions are obtained:

x=b±b24ac2ax={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}


1+52=1.618033988749894848204586834365{\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,749\,894\,848\,204\,586\,834\,365\dots

and

152=0.6180339887{\frac {1-{\sqrt {5}}}{2}}=-0.618\,033\,988\,7\dots


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

φ=1+52=1.618033988749894848204586834365\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,749\,894\,848\,204\,586\,834\,365\dots


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