Question

what is golden ratio? Prove that it is irrational

Answer 1

Answer:

I am struggling to see where the contradiction lies in my proof. In a previous example, 1/ϕ=ϕ−11/ϕ=ϕ−1 where ϕϕ is the golden ratio 5√+125+12.

Since I am proving by contradiction, I started out by assuming that ϕϕ is rational. Then, by definition, there exists a,ba,b such that ϕ=a/bϕ=a/b. After some simple calculations and using the result shown from my previous example, I found that ϕ=b/(a−b)ϕ=b/(a−b). I also know that b<ab<a from directly calculating the ratio.

I know there is a contradiction in the result ϕ=b/(a−b)ϕ=b/(a−b) but I cannot see it. Any help would be appreciated