Since the

golden ratio satisfies this

equation, we can immediately deduce its

continued fraction. Let

φ be the golden ratio. We want the positive

solution. So φ-1 = 1/φ > 0, so φ > 1. But this means φ-1 = 1/φ < 1, so 1 < φ < 2, and the first term in the continued fraction of φ is

1.

So now we want to write φ = 1 + 1/`x`. But this means `x` = 1/(φ-1) = φ, so φ = 1 + 1/φ. Expanding, we write

1
φ = 1 + ---------------------
1
1 + -----------------
1
1 + -------------
1
1 + ---------
1 + ...