# Definition:Absolute Value

## Contents

## Definition

### Definition 1

Let $x \in \R$ be a real number.

The **absolute value** of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows:

- $\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

### Definition 2

Let $x \in \R$ be a real number.

The **absolute value** of $x$ is denoted $\size x$, and is defined as:

- $\size x = +\sqrt {x^2}$

where $+\sqrt {x^2}$ is the positive square root of $x^2$.

## Graphical Illustration

The graph of the **absolute value function** can be presented as:

## Number Classes

The absolute value function applies to the various number classes as follows:

- Natural numbers $\N$: All elements of $\N$ are greater than or equal to zero, so the concept is irrelevant.
- Integers $\Z$: As defined here.
- Rational numbers $\Q$: As defined here.
- Real numbers $\R$: As defined here.

- Complex numbers $\C$: As $\C$ is not an ordered set, the definition of the absolute value function based upon whether a complex number is greater than or less than zero cannot be applied.

The notation $\cmod z$, where $z \in \C$, is defined as the modulus of $z$ and has a different meaning.

## Ordered Integral Domain

We can go still further back, and consider the general ordered integral domain:

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$.

Then for all $a \in D$, the **absolute value** of $a$ is defined as:

- $\size a = \begin{cases} a & : 0_D \le a \\ -a & : a < 0_D \end{cases}$

## Also known as

The **absolute value of $x$** is sometimes called the **modulus** or **magnitude** of $x$, but note that modulus has a more specialized definition in the domain of complex numbers, and that magnitude has a more specialized definition in the context of vectors.

Some sources refer to it as the **size of $x$**.

Some sources call it the **numerical value**.

## Examples

### Absolute Value of $3$ and $-3$

- $\size 3 = 3 = \size {-3}$

### Absolute Value of $-6$

- $\size {-6} = 6$

### Absolute Value of $3 - 5$

- $\size {3 - 5} = \size {5 - 3} = 2$

### Absolute Value of $x - a$

Let $x, a \in \R$.

Then:

- $\size {x - a} = \begin {cases} x - a & : x \ge a \\ a - x & : x < a \end {cases}$

### Absolute Value of $0$

- $\size 0 = 0$

## Also see

- Results about
**the absolute value function**can be found here.

### Generalizations

## Technical Note

$\mathsf{Pr} \infty \mathsf{fWiki}$ has a $\LaTeX$ shortcut for the symbol used to denote absolute value:

- The $\LaTeX$ code for \(\size {x}\) is
`\size {x}`

.

If the argument of the `\size`

command is $1$ character, then the braces `{}`

are usually omitted.

## Sources

- 2003: John H. Conway and Derek A. Smith:
*On Quaternions And Octonions*... (previous) ... (next): $\S 1$: The Complex Numbers: Introduction: $1.1$: The Algebra $\R$ of Real Numbers