Definition:Absolute Value

Definition

Let $x$ be a number.

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

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

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

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

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 $\left\vert{z}\right\vert$, where $z \in \C$, is defined as the modulus of $z$ and has a different meaning.

On $\Q$ and $\R$, the absolute value defines a norm, as proved in Absolute Value is Norm.

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 (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 $\left\vert{x}\right\vert$ can also be defined as $\left\vert{x}\right\vert = \sqrt {x^2}$.

Ordered Integral Domain

We can go still further back, and consider the general ordered integral domain $\left({D, +, \times}\right)$ whose ordering induced by the postivity property is $\le$.

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

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

It is clear that the definition for numbers is compatible with this, from:

• Integers form Ordered Integral Domain
• Rational Numbers form Ordered Integral Domain
• Real Numbers form Ordered Integral Domain

Abstract Absolute Value

For any field $\left({k, +, \cdot}\right)$, a norm on $k$ is also referred to as an absolute value on $k$.

Sources

• James M. Hyslop: Infinite Series (1942): $\S 4$: Footnote
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 2.7$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Exercise $1$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.14$