# Symbols:Greek/Phi

## Contents

## Phi

The $21$st letter of the Greek alphabet.

- Minuscules: $\phi$ and $\varphi$

- Majuscules: $\Phi$ and $\varPhi$

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

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The $\LaTeX$ code for \(\varphi\) is `\varphi`

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The $\LaTeX$ code for \(\Phi\) is `\Phi`

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The $\LaTeX$ code for \(\varPhi\) is `\varPhi`

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## 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`

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## 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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