# Proof that φ is irrational

219,312pages on
this wiki

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.