φ (phi) is the number such that is the positive solution to the equation (defined implicitly) A proof of its irrationality: 1. * (given) 2. * (subtract phi and 1 from both sides) 3. * (Quadratic formula) 4. * (Simplify) 5. * (Positive root only) 6. * (Separate fractions) 7. * Since the square root of 5 is irrational, and an irrational number divided by a rational number is an irrational number, and an irrational number plus a rational number is an irrational number, φ is irrational.