φ (phi) is the number such that is the positive solution to the equation (defined implicitly) \phi^2 = \phi + 1

A proof of its irrationality:

  1. \phi^2 = \phi + 1 (given)
  2. \phi^2 - \phi - 1 = 0 (subtract phi and 1 from both sides)
  3. \phi = \frac{1 \pm \sqrt {(-1)^2-4(1)(-1)}}{2(1)}, (Quadratic formula)
  4. \phi = \frac{1 \pm \sqrt {5}}{2}, (Simplify)
  5. \phi = \frac{1 + \sqrt{5}}{2} (Positive root only)
  6. \phi = \frac{1}{2} + \frac{\sqrt{5}}{2} (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.

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