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==Calculation==
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As discussed, the golden ratio is a ratio such that
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<center> <math> \frac{a+b}{a} = \frac{a}{b} </math> </center>
 
<center> <math> \frac{a+b}{a} = \frac{a}{b} </math> </center>
  

Revision as of 17:42, 2 December 2018

Calculation

As discussed, the golden ratio is a ratio such that

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

We can solve this equation to find an explicit quantity for the ratio.

$ LHS = \frac{a}{b} + \frac{b}{a} = 1 + \frac{b}{a} $
$ 1 + \frac{b}{a} = \frac{a}{b} $

We set the ratio equal to a certain quantity given by r.

$ r = \frac{a}{b} $

Then we can solve for the ratio numerically.

$ 1 + \frac{1}{r} = r $
$ r + 1 = r^2 $

We can see from the above result that the golden ratio can also be described as a ratio such that in order to get the square of the ratio, you add one to the ratio.

$ r^2 - r - 1 = 0 $

We can then apply the quadratic formula to solve for the roots of the equation.

$ r = \frac{1 \pm \sqrt{1^2 - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2} $

The positive root is then the golden ratio.

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