Proof that φ is irrational

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, $$\phi \!\ $$ is irrational.