Two quantities a and b are said to be in the golden ratio φ if
aa+b=ba=φ
One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting inab=φ1,
aa+b=aa+ab=1+ab=1+φ1 b/a = 1/φ,
1+φ1=φ
φ+1=φ2
φ2−φ−1=0.
Using the quadratic formula, two solutions are obtained:
x=2a−b±b2−4ac
21+5=1.618033988749894848204586834365…
and
21−5=−0.6180339887…
Because φ is the ratio between positive quantities, φ is necessarily positive:
φ=21+5=1.618033988749894848204586834365…
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