The Golden Ratio Part 2 – Algebraic Properties

The Golden Ratio, has its origins in early geometry when the Classical Greeks noticed that constant embedded in the pentagram and the golden rectangle. Due to the nature of where it was found, Golden, defined as \phi = \frac{1+\sqrt{5}}{2}, has a few interesting properties.

\phi is the only number such that \phi^2 = \phi + 1. The Golden Ratio is also related to the Fibonacci sequence through raising \phi to a power.

  1. \phi ^1 = \phi
  2. \phi ^2 = \phi + 1
  3. \phi ^3 = 2 \phi +1
  4. \phi ^4 = 3 \phi + 2
  5. \phi ^5 = 5 \phi + 3

Notice while the powers of \phi on the left hand side are indexes of the Fibonacci numbers, the coefficients of \phi on the right hand side are the corresponding Fibonacci Numbers.

Another interesting fact is that 1/ \phi = \phi - 1. Thus \phi ( 1 / \phi ) = \phi ( \phi - 1) = \phi ^ 2 - \phi = \phi +1 - \phi = 1, or you could just cancel out the \phi‘s.