# Symbols:Z

### zepto-

$\mathrm z$

The Système Internationale d'Unités symbol for the metric scaling prefix zepto, denoting $10^{\, -21 }$, is $\mathrm { z }$.

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

### zetta-

$\mathrm Z$

The Système Internationale d'Unités symbol for the metric scaling prefix zetta, denoting $10^{\, 21 }$, is $\mathrm { Z }$.

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

### Variable

$z$

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

### Complex Variable

$z$

Used to denote a general variable in the complex plane.

The $\LaTeX$ code for $z$ is z .

### Zenith Distance

$z$

Let $X$ be the position of a star (or other celestial body) on the celestial sphere.

The zenith distance $\zeta$ of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between $X$ and the zenith.

It is often denoted by $z$.

### 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 .

### Set of Integers

$\Z$

The set of integers:

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

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

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

### Set of Non-Zero Integers

$\Z_{\ne 0}$

The set of non-zero integers:

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

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

### Set of Non-Negative Integers

$\Z_{\ge 0}$

The set of non-negative integers:

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

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

### Set of Strictly Positive Integers

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

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

### Set of Integers Modulo m

$\Z_m$

Let $m \in \Z$ be an integer.

The integers modulo $m$ are the set of least positive residues of the set of residue classes modulo $m$:

$\Z_m = \set {0, 1, \ldots, m - 1}$

The $\LaTeX$ code for $\Z_m$ is \Z_m  or \mathbb Z_m or \Bbb Z_m.

### Reduced Residue System

$\Z'_m$

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

$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$

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

$\Z'_m = \set {\eqclass {a_1} m, \eqclass {a_2} m, \ldots, \eqclass {a_{\map \phi m} } m}$

where:

$\forall k: a_k \perp m$
$\map \phi m$ denotes the Euler phi function of $m$.

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

### Set of Integer Multiples

$n \Z$

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

$\set {x \in \Z: n \divides x}$

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 .

### Set of Gaussian Integers

$\Z \sqbrk i$

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 $\Z \sqbrk i$, and hence can be defined as:

$\Z \sqbrk i = \set {a + b i: a, b \in \Z}$

The $\LaTeX$ code for $\Z \sqbrk i$ is \Z \sqbrk i .

### Initial Segment of Natural Numbers

$\map \Z n$

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

$\map \Z n = \set {x \in \Z: 1 \le x \le n} = \set {1, 2, \ldots, n}$

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

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

### Zermelo-Fraenkel Set Theory

ZF

An abbreviation for Zermelo-Fraenkel Set Theory, a system of axiomatic set theory upon which most of conventional mathematics can be based.

### Zermelo-Fraenkel Set Theory with the Axiom of Choice

ZFC

An abbreviation for Zermelo-Fraenkel Set Theory with the Axiom of Choice, a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based.