Continued Fractions

Write 1 + 1/(1 + 1/(1 + 1/(1 + …))) — a fraction nested inside itself forever. What number is it? Remarkably, it's the golden ratio φ ≈ 1.618. This is a continued fraction, and it reveals something the decimal expansion hides: φ is the 'most irrational' number of all, the hardest to approximate by fractions. Continued fractions expose a number's true approximability.