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 . But since we want the ratio to be positive . Now we want to know what the difference between the golden ratio and one over the golden ratio is:
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.