φ (phi) is the number such that is the positive solution to the equation (defined implicitly)
A proof of its irrationality:
- (subtract phi and 1 from both sides)
- (Quadratic formula)
- (Positive root only)
- (Separate fractions)
- 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.