How to Evaluate the Golden Ratio as a Continued Fraction

Updated 12 October 2026

Evaluate an infinite nested fraction

x = 1 + 1 / (1 + 1 / (1 + ...)) = $\varphi \left(1.618\dots \right)$φ (1.618...)

What does the infinite fraction 1 + 1/(1 + 1/(1 + …)) equal?

The nested fraction 1 + 1/(1 + 1/(1 + …)) equals the golden ratio, (1 + √5) ÷ 2, which is about 1.618. The key is that the tail repeats the whole expression. Call the value x. Everything under the first fraction bar is x again, so x = 1 + 1/x. Multiply through by x to get x² = x + 1, which rearranges to x² − x − 1 = 0. The quadratic formula gives x = (1 ± √5) ÷ 2, and only the positive root can be the value, because every part of the expression is positive. You can watch it converge by cutting the fraction off at each depth: 1, then 2, then 1.5, then about 1.667, then 1.6, then 1.625. Each value is 1 plus one over the one before, and they land alternately below and above 1.618. They are the ratios of neighboring Fibonacci numbers: 2/1, 3/2, 5/3, 8/5 and 13/8.

1. The denominator repeats the whole expression, so the tail is x again.

2. Multiply through by x: $x^2=x+1$x² = x + 1, which is $x^2-x-1=0$x² − x − 1 = 0.

3. The positive root is $\left(1+\sqrt{5}\right)÷2$(1 + √5) ÷ 2, about 1.618 — the golden ratio.

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