Phi

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Phi (in lowercase, either $\phi$ or $\varphi$ ; capitalized, $\Phi$ ) is the 21st letter in the Greek alphabet. It is used frequently in mathematical writing, often to represent the constant $\frac{1+\sqrt{5}}{2}$ . (The Greek letter tau ( $\tau$ ) was also used for this purpose in pre-Renaissance times.)

Use

$\phi$ 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 $x^2-x-1=0$ .

$\phi$ is also equal to the continued fraction $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}$ and the continued radical $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$ . It is the only positive real number that is one more than its multiplicative inverse.

It is also ${\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}$ where $F_n$ is the nth number in the Fibonacci sequence.

Golden ratio

$\phi$ 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 $\phi$ ; it has a number of interesting properties.

The first fifteen digits of $\phi$ in decimal representation are $1.61803398874989$

Other Usages

$\phi$ is also commonly used to represent Euler's totient function.

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Phi

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Use

Golden ratio

Other Usages

See also