Mathematical Properties of the Golden Ratio

The Golden Ratio, Φ, is an irrational number that has the following unique properties:

  1. Taking the reciprocal of Φ and adding one yields Φ. phi=1/phi+1, or Φ=1/Φ+1.
  2. Φ squared equals itself plus one. In other words, Phi^2 =Phi+1, or Φ^2=Φ+1. These characteristics are indeed very interesting; it is the only number in the world has such properties.
  3. If we convert the equation from 2. into the equation Φ^2-Φ-1=0 which is in the format ax^2 + bx + c = 0, so we can solve using the quadratic formula, x= (-b ± √(b^2 - 4ac))/(2a). Doing this we get x = (1 ± √5)/2. Together, these two solutions are known as Phi (1.618033989) and phi (0.618033989). Phi and phi are reciprocals.

Φ vs. Π

Φ and Π (pi) have this in common: where Π is the ratio of the circumference to its diameter, Φ is the ratio of the length to the width of a perfect rectangle.

 

Valid XHTML 1.0!