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Introduction

The golden ratio is a ratio such that, given two quantities a and b,

$ \frac{a+b}{a} = \frac{a}{b} $
$ LHS = \frac{a}{b} + \frac{b}{a} = 1 + \frac{b}{a} $
$ 1 + \frac{b}{a} = \frac{a}{b} $
$ r = \frac{a}{b} $
$ 1 + \frac{1}{r} = r $
$ r + 1 = r^2 $
$ r^2 - r - 1 = 0 $
$ r = \frac{1 \pm \sqrt{1^2 - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2} $
$ \frac{1 \pm \sqrt{5}}{2} = \phi $

MathIsFun also has an interactive display that can construct a rectangle in the golden ratio given a certain fixed width or length.

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