# 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 $2\sin\frac{\pi}{10}$