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.)


$\phi$ 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 square and one less than its reciprocal.

It is also ${\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}$ where $F_n$ is the nth number in the Fibonacci sequence. In other words, if you divide the $n$th term of the Fibonacci series over the $(n-1)$th term, the result approaches $\phi$ as $n$ increases.

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

See also