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The positive root is then the golden ratio.
 
The positive root is then the golden ratio.
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<center> <math> r = \frac{1 + \sqrt{5}}{2} = 1.61803398875</math> </center>
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[[Walther MA279 Fall2018 topic2|Back to Home]]
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<br><br>[[Category:MA279Fall2018Walther]]

Latest revision as of 16:47, 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.

$ r = \frac{1 + \sqrt{5}}{2} = 1.61803398875 $

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