The golden ratio φ satisfies the following equation.
The proof most commonly given is to let x equal the right-hand side of the equation, then observe that x² = 1 + x, the quadratic equation for the golden ratio. The quadratic has two roots: φ and −1/φ. Since x > 1, x = φ.
This proof tacitly assumes that the expression above is meaningfully defined, and then manipulates it algebraically. But is it meaningfully defined? What exactly does the infinitely nested sequence of radicals mean?
You could interpret the nested radicals to be the limit of the iteration
and show that the limit exists.
What should the starting value of our iteration be? It seems natural to choose x = 1, but other values will do. More on that shortly. We could compute φ by implementing the iteration in Python as follows.
x_old, x_new = 0, 1
while (abs(x_old - x_new) > 1e-15):
x_new, x_old = (1 + x_new)**0.5, x_new
print(x_new)
The program terminates, so the iteration must converge, right? Probably so, but that’s not a proof. To be more rigorous, define
and so
This shows 0 < f ′(x) < ½ for any positive x and so the function f(x) is a contraction mapping. This means the iterations converge. We could start with any x > −¾ and the derivative would still be less than 1, so the iterations would converge.
We can visualize the process of convergence starting with x = 0 using the following cobweb plot.
