φ (phi) is the positive solution to x² = x + 1, giving (1+√5)/2 ≈ 1.61803. It is the limit of consecutive Fibonacci numbers: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8… converges to φ. This makes φ the 'most irrational' number: hardest to approximate by fractions.
The golden ratio has been celebrated in art and architecture since antiquity. Whether it genuinely appears in the Parthenon or Da Vinci's paintings is debated, but it definitively appears in pentagons and pentagrams, in phyllotaxis (leaf and seed arrangements), and in the geometry of quasicrystals.
φ satisfies φ² = φ + 1 and 1/φ = φ − 1. Its continued fraction [1; 1, 1, 1, …] is the simplest possible: all 1s. Its decimal expansion is 1.6180339887…