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Golden ratio

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The golden ratio is an irrational number equal to \phi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots. It has the special property of being one more than its reciprocal. Equivalently, two positive real numbers a > b are in the golden ratio if the ratio of a + b to a is the same as the ratio of a to b.

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 \phi:

  • \frac{1+\sqrt{5}}{2}
  • 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ._{._.}}}}} (continued fraction)
  • 1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \ldots}}} (nested radical)
  • 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \ldots (sums of the reciprocals of the Fibonacci numbers)
  • 2\sin\frac{3\pi}{10}
  • The ratio between the side length of a regular pentagon and one of its diagonals.

A capital phi (\Phi) denotes the reciprocal of \phi. It is equal to 2\sin\frac{\pi}{10}


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