# Difference between revisions of "Phi"

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<math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the [[Fibonacci sequence]], as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>. | <math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the [[Fibonacci sequence]], as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>. | ||

− | <math>\phi</math> is also equal to the [[continued fraction]] <math>1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}</math> and the [[continued radical]] <math>\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}</math>. It is the only positive real number that is one more than its [[multiplicative inverse]]. | + | <math>\phi</math> is also equal to the [[continued fraction]] <math>1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}</math> and the [[nested radical|continued radical]] <math>\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}</math>. It is the only positive real number that is one more than its [[multiplicative inverse]]. |

It is also <math>{\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}</math> where <math>F_n</math> is the nth number in the [[Fibonacci sequence]]. | It is also <math>{\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}</math> where <math>F_n</math> is the nth number in the [[Fibonacci sequence]]. |

## Revision as of 16:38, 10 March 2014

**Phi** (in lowercase, either or ; capitalized, ) is the 21st letter in the Greek alphabet. It is used frequently in mathematical writing, often to represent the constant . (The Greek letter Tau () was also used for this purpose in pre-Renaissance times.)

## Contents

## Use

appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the Fibonacci sequence, as well as the positive solution of the quadratic equation .

is also equal to the continued fraction and the continued radical . It is the only positive real number that is one more than its multiplicative inverse.

It is also where is the nth number in the Fibonacci sequence.

## Golden ratio

is also known as the Golden Ratio. It was commonly believed by the Greeks to be the most aesthetically pleasing ratio between side lengths in a rectangle. The Golden Rectangle is a rectangle with side lengths of 1 and ; it has a number of interesting properties.

The first fifteen digits of in decimal representation are

## Other Usages

- is also commonly used to represent Euler's totient function.