# Symbols:Z

## zepto-

$\mathrm z$

The Système Internationale d'Unités metric scaling prefix denoting $10^{\, -21 }$.

Its $\LaTeX$ code is \mathrm {z} .

## zetta-

$\mathrm Z$

The Système Internationale d'Unités metric scaling prefix denoting $10^{\, 21 }$.

Its $\LaTeX$ code is \mathrm {Z} .

## General Variable

$z$

Used to denote a general variable, usually in conjunction with other variables $x$ and $y$.

### Complex Variable

Used to denote a general variable in the complex plane.

The $\LaTeX$ code for $z$ is z .

## Random Variable

$Z$

Used to denote a general random variable, usually in conjunction with another random variables $X$ and $Y$.

The $\LaTeX$ code for $Z$ is Z .

## The Set of Integers

$\Z$

The set of integers:

$\Z = \left\{{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}\right\}$.

From the German Zahlen, which means (whole) numbers.

Its $\LaTeX$ code is \Z  or \mathbb Z or \Bbb Z.

## The Set of Non-Zero Integers

$\Z_{\ne 0}$

The set of non-zero integers:

$\Z_{\ne 0} = \Z \setminus \left\{{0}\right\} = \left\{{\ldots, -3, -2, -1, 1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z_{\ne 0}$ is \Z_{\ne 0} .

### Deprecated

$\Z^*$

The set of non-zero integers:

$\Z^* = \Z \setminus \left\{{0}\right\} = \left\{{\ldots, -3, -2, -1, 1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z^*$ is \Z^* .

## The Set of Non-Negative Integers

$\Z_{\ge 0}$

The set of non-negative integers:

$\Z_{\ge 0} = \left\{{n \in \Z: n \ge 0}\right\} = \left\{{0, 1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z_{\ge 0}$ is \Z_{\ge 0}  or \mathbb Z_{\ge 0}.

### Deprecated

$\Z_+$

The set of non-negative integers:

$\Z_+ = \left\{{n \in \Z: n \ge 0}\right\} = \left\{{0, 1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z_+$ is \Z_+  or \mathbb Z_+.

## The Set of Strictly Positive Integers

$\Z_{> 0}$
$\Z_{> 0} = \left\{{n \in \Z: n > 0}\right\} = \left\{{1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z_{> 0}$ is \Z_{> 0}  or \mathbb Z_{> 0} or \Bbb Z_{> 0}.

### Deprecated

$\Z_+^*$
$\Z_+^* = \left\{{n \in \Z: n > 0}\right\} = \left\{{1, 2, 3, \ldots}\right\}$

The $\LaTeX$ code for $\Z_+^*$ is \Z_+^*  or \mathbb Z_+^* or \Bbb Z_+^*.

## Reduced Residue System Modulo $m$

$\Z'_m$

The reduced residue system modulo $m$, denoted $\Z'_m$, is the set of all residue classes of $k$ (modulo $m$) which are prime to $m$:

$\Z'_m = \left\{{\left[\!\left[{k}\right]\!\right]_m \in \Z_m: k \perp m}\right\}$

Thus $\Z'_m$ is the set of all coprime residue classes modulo $m$:

$\Z'_m = \left\{ {\left[\!\left[{a_1}\right]\!\right]_m, \left[\!\left[{a_2}\right]\!\right]_m, \ldots, \left[\!\left[{a_{\phi \left({m}\right)} }\right]\!\right]_m}\right\}$

where:

$\forall k: a_k \perp m$
$\phi \left({m}\right)$ denotes the Euler phi function of $m$.

The $\LaTeX$ code for $\Z'_m$ is \Z'_m  or \mathbb Z'_m or \Bbb Z'_m.

## The Set of Integer Multiples

$n \Z$

The Set of Integer Multiples $n \Z$ is defined as:

$\left\{ {x \in \Z: n \mathrel \backslash x}\right\}$

for some $n \in \N$.

That is, it is the set of all integers which are divisible by $n$, that is, all multiples of $n$.

The $\LaTeX$ code for $n \Z$ is n \Z .

## The Gaussian Integers

$\Z \left[{i}\right]$

A Gaussian integer is a complex number whose real and imaginary parts are both integers.

That is, a Gaussian integer is a number in the form:

$a + b i: a, b \in \Z$

The set of all Gaussian integers can be denoted $\Z \left[{i}\right]$, and hence can be defined as:

$\Z \left[{i}\right] = \left\{{a + b i: a, b \in \Z}\right\}$

The $\LaTeX$ code for $\Z \left[{i}\right]$ is \Z \left[{i}\right] .

## Subsets of Integers

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

Used by some authors to denote the set of all integers between $1$ and $n$ inclusive:

$\Z \left({n}\right) = \left\{{x \in \Z: 1 \le x \le n}\right\} = \left\{{1, 2, \ldots, n}\right\}$

That is, an alternative to Initial Segment of Natural Numbers $\N^*_n$.

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