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The golden ratio is an irrational number equal to . It is one of the two solutions of the quadratic equation: It has the special property of being one more than its reciprocal. Equivalently, two positive real numbers are in the golden ratio if the ratio of to is the same as the ratio of to .

The golden ratio has many mysterious reappearances in geometry and number theory. In particular it has a close relationship to the number 5 and the Fibonacci numbers. It also has many applications in art and architecture.

There are several formulas and definitions for :

  • (continued fraction)
  • (nested radical)
  • (sums of the reciprocals of the Fibonacci numbers)
  • The ratio between the side length of a regular pentagon and one of its diagonals.

A capital phi () denotes the reciprocal of . It is equal to

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