# Symbols:Greek/Phi

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## Phi

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

$\map \phi n$

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:

$\map \phi n =$ the number of strictly positive integers less than or equal to $n$ which are prime to $n$

That is:

$\map \phi n = \card {S_n}: S_n = \set {k: 1 \le k \le n, k \perp n}$

The $\LaTeX$ code for $\map \phi n$ is \map \phi n .

### Golden Mean

$\phi$

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

$AB : AC = AC : BC$

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

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

### Mapping

$\map \phi x$

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 $\map \phi x$ is \map \phi x .

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