Question

I will love any input on this problem. Thank you.
1. The Golden Ratio has a long history for its real occurrence in nature and for the supposed mystical properties that various groups have attributed to it. In other words, it is the ratio of the sides of a rectangle that, when a square is cut from it, leaves another smaller rectangle, the sides of which have the same ratio as the original rectangle. We are told that the Golden Ratio is irrational. Prove that the Golden Ratio must be irrational.

Answer 1

What is the actual value of the golden ratio, call it r? It is
\[ r = \frac{\sqrt{5} +1}{2} \]

This number is irrational because \( \sqrt{5} \) is irrational. So if you want to prove r is irrational, you need to show that \( \sqrt{5} \) is irrational.

Answer 2

Suppose the Golden Ratio is a rational number p/q, which is in its lowest terms, that is, the highest common factor of p and q is 1.



Let's assume that b=1. This can be done since the rectangle can be of any size as long as the Golden Ratio is satisfied.

We have that p/q = a/1 = 1/(a-1)
Hence a(a-1) = 1
a^2-a-1=0
a^2 = a+1
But a = p/q, and hence

(p/q)^2 = (p/q)+1
p^2 / q^2 = (p+q)/q
p^2 / q = p+q
(p/q) = (p+q)/p
p/q = 1 + q/p

(p/q) - (q/p) = 1

This is impossible; the difference between a fraction and its reciprocal can never be 1. Hence the Golden Ratio cannot be rational; thus it is irrational.