From ProofWiki
Jump to: navigation, search

Previous  ... Next


The $21$st letter of the Greek alphabet.

Minuscules: $\phi$ and $\varphi$
Majuscules: $\Phi$ and $\varPhi$

The $\LaTeX$ code for \(\phi\) is \phi .
The $\LaTeX$ code for \(\varphi\) is \varphi .

The $\LaTeX$ code for \(\Phi\) is \Phi .
The $\LaTeX$ code for \(\varPhi\) is \varPhi .

Euler Phi Function

$\phi \left({n}\right)$

Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The Euler $\phi$ (phi) function is the arithmetic function $\phi: \Z_{>0} \to \Z_{>0}$ defined as:

$\phi \left({n}\right) = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$

That is:

$\phi \left({n}\right) = \left|{S_n}\right|: S_n = \left\{{k: 1 \le k \le n, k \perp n}\right\}$

The $\LaTeX$ code for \(\phi \left({n}\right)\) is \phi \left({n}\right) .

Golden Mean


Let a line segment $AB$ be divided at $C$ such that:

$AB : AC = AC : BC$

Then the golden mean $\phi$ is defined as:

$\phi := \dfrac {AB} {AC}$

The $\LaTeX$ code for \(\phi\) is \phi .


$\phi \left({x}\right)$

The Greek letter $\phi$, along with $\psi$ and $\chi$ and others, is often used to denote a general mapping.

In the context of abstract algebra, it often denotes a homomorphism.

The $\LaTeX$ code for \(\phi \left({x}\right)\) is \phi \left({x}\right) .