Definition:Absolute Value

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Definition

Definition 1

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


The absolute value of $x$ is denoted $\left\vert{x}\right\vert$, and is defined using the ordering on the real numbers as follows:

$\left\vert{x}\right\vert = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$


Definition 2

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

The absolute value of $x$ is denoted $\left\vert{x}\right\vert$, and is defined as:

$\left\vert{x}\right\vert = +\sqrt {x^2}$

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


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 $\left\vert{z}\right\vert$, 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 $\left({D, +, \times}\right)$ be an ordered integral domain whose ordering induced by the postivity property is $\le$.

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

$\left\vert{a}\right\vert = \begin{cases} a & : 0 \le a \\ -a & : a < 0 \end{cases}$


Abstract Absolute Value

Let $\struct {\mathbb k, +, \cdot}$ be a field.

A norm on $k$ is also referred to as an absolute value on $k$.


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.


Also see

  • Results about the absolute value function can be found here.


Generalizations