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 $\left|{x}\right| = \sqrt {x^2}$.

Abstract absolute value
For any field $k$, an absolute value on $(k,+,\cdot)$ is a function $|\cdot | : k \to \R$ satisfying: for all $x,y \in k$. The pair $\left(k,|\cdot |\right)$ is called a valued field.
 * $|x| \geq 0$ and $|x|=0 \Leftrightarrow x = 0$
 * $|x\cdot y|=|x|\cdot |y|$
 * $|x + y| \leq |x| + |y|$