Definition:Absolute Value

Definition
Let $$x$$ be a number.

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



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

Note that since $$0 = -0$$, the value of $$\left|{x}\right|$$ at $$x = 0$$ is often included in one of the other two cases, most commonly:

\left|{x}\right| = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases} $$ but this can be argued as being less symmetrically aesthetic.

Note that the absolute value is functional.

It 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 concept as defined here can not be applied. The notation $$\left|{z}\right|$$, where $$z \in \C$$, is defined as the modulus of $$z$$ and has a different meaning.

This is sometimes called the modulus or magnitude of $$x$$, but note that modulus has a more specialized definition in the domain of complex numbers (see above), and that magnitude has a more specialized definition in reference to vectors (see above).

From Even Powers are Positive, it can be seen that $$\left|{x}\right|$$ can also be defined as $$\left|{x}\right| = \sqrt {x^2}$$.