Golden Ratio

Def. a and b are in the golden ratio if a/b=b/a-b. The symbolScreen Shot 2014-05-29 at 12.41.51 PM is used for the golden ratio.

Lets calculate the golden ratio. To do this let b=1. We have a/1=1/a-1. Therefor a^2-a-1=0. If we solve this quadratic equation we get Screen Shot 2014-05-29 at 12.25.01 PM. But since we want the ratio to be positive Screen Shot 2014-05-29 at 12.25.01 PM. Now we want to know what the difference between the golden ratio and one over the golden ratio is:

Screen Shot 2014-05-29 at 12.37.49 PM So x is equal to 1.

It follows:

Screen Shot 2014-05-29 at 12.46.35 PM

This is a very famous formula.

 

Proof of the irrationality of the golden ratio by contradiction:

Lets assume that the golden ratio is rational. Then it can be written as x/y when x and y are integers. We assume that the x and y are the smallest numbers where x/y is the golden ratio. If x and y are integers then y and x-y are integers to. Per definition y/(x-y) is also the golden ratio. This is a contradiction.

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