How to Evaluate the Golden Ratio as a Continued Fraction

Evaluate an infinite nested fraction

Lex the Fox

x=1+11+11+1⋱x = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ddots}}} = φ (1.618...)

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

    the tail repeats the whole

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

    multiply through by x

  3. The positive root is (1 + √5) ÷ 2, about 1.618 — the golden ratio.

    the positive root

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.

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Value of 1 + 1/(1 + 1/(1+...)) = ?

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