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 $\size x$, 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:


Absolute-value-function.png


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.

Some call it just the value, but that term is too broad to be reliable.


Examples

Absolute Value of $3$ and $-3$

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


Absolute Value of $-2$

$\size {-2} = 2$


Absolute Value of $-6$

$\size {-6} = 6 = \size 6$


Absolute Value of $\dfrac 3 4$

$\size {\dfrac 3 4} = \dfrac 3 4$


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 $\size x \le 2$

$\size x \le 2 \iff -2 \le x \le 2$


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

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