In mathematics, the **absolute value** or **modulus** of a real number x, denoted |*x*|, is the non-negative value of x without regard to its sign. Namely, |*x*| = *x* if x is positive, and |*x*| = −*x* if x is negative (in which case −*x* is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

- 1 Terminology and notation
- 2 Definition and properties
- 3 Absolute value function
- 4 Distance
- 5 Generalizations
- 5.1 In mathematics, the
**absolute value**or**modulus**of a real number x, denoted |*x*|, is the non-negative value of x without regard to its sign. Namely, |*x*| =*x*if x is positive, and |*x*| = −*x*if x is negative (in which case −*x*is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

## Contents

- 1 Terminology and notation
- 2 Definition and properties
- 2.1 Real numbers
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

In 1806, Jean-Robert Argand introduced the term

*module*, meaning*unit of measure*in French, specifically for the*complex*absolute value,^{[1]}^{[2]}and it was borrowed into English in 1866 as the Latin equivalent*modulus*.^{[1]}The term*absolute value*has been used in this sense from at least 1806 in French^{[3]}and 1857 in English.^{[4]}The notation |*x*|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.^{[5]}Other names for*absolute value*include*numerical value*^{[1]}and*magnitude*.^{[1]}In programming languages and computational software packages, the absolute value of*x*is generally represented by`abs(`

, or a similar expression.*x*)The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the euclidean norm

^{[6]}or sup norm^{[7]}of a vector in $\mathbb {R} ^{n}$, although double vertical bars with subscripts ($||\cdot ||_{2}$ and $$$\mathbb {R} ^{n}$, although double vertical bars with subscripts ($||\cdot ||_{2}$ and $||\cdot ||_{\infty }$, respectively) are a more common and less ambiguous notation.For any real number x, the

**absolute value**or**modulus**of x is denoted by |*x*| (a vertical bar on each side of the quantity) and is defined as^{[8]}- $$
The absolute value of x is thus always either positive or zero, but never negative: when x itself is negative (

*x*< 0), then its absolute value is necessarily positive (|*x*| = −*x*> 0).From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below).

Since the square root symbol represents the unique

*positive*square root (when applied to a positive number), it follows that- $|x|={\sqrt {x^{2}}}$

is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.

^{[9]}The absolute value has the fo

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below).

Since the square root symbol represents the unique

*positive*square root (when applied to a positive number), it follows thatis equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.

^{[9]}The absolute value has the following four fundamental properties (

*a*,*b*are real numbers), that are used for generalization of this notion to other domains:The norm of a vector is also called its

*length*or*magnitude*.In the case of Euclidean space

**R**^{n}, the function defined by- $\|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}$

is a norm called the Euclidean norm. When the real numbers

**R**are considered as the one-dimensional vector space**R**^{1}, the absolute value is a norm, and is the p-norm (see L^{p}space) for any p. In fact the absolute value is the "only" norm on**R**^{1}, in the sense that, for every norm || · || on**R**^{1}, ||*x*|| = ||1|| ⋅ |*x*|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the EAgain the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a vector space V over a field F, represented as || · ||, is called an

**absolute value**, but more usually a**norm**, if it satisfiesA real-valued function on a vector space V over a field F, represented as || · ||, is called an

**absolute value**, but more usually a**norm**, if it satisfies the following axioms:For all a in F, and

**v**,**u**in V,The norm of a vector is also called its

*length*or*magnitude*.In the case of Euclidean space

**R**^{n}, the function defined by- $\|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}$

is a norm called the Euclidean norm. When the real numbers

**R**are considered as the one-dimensional vector space**R**^{1}, the absolute value is a norm, and is the p-norm (see L^{p}space) for any p. In fact the absolute value is the "only" norm on**R**^{1}, in the sense that, for every norm || · || on**R**^{1}, ||*x*|| = ||1|| ⋅ |*x*|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane**R**^{2}.### Composition algebras

Main article: Composition algebraEvery composition algebra

*A*has an involution*x*→*x** called its**conjugation**. The product in*A*of an element*x*and its conjugate*x** is written*N*(*x*) =*x x** and called the**norm of x**.The real numbers ℝ, complex numbers ℂ, and quaternions ℍ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However,

In the case of Euclidean space

**R**^{n}, the function defined byis a norm called the Euclidean norm. When the real numbers

**R**are considered as the one-dimensional vector space**R**^{1}, the absolute value is a norm, and is the p-norm (see L^{p}space) for any p. In fact the absolute value is the "only" norm on**R**^{1}, in the sense that, for every norm || · || on**R**^{1}, ||*x*|| = ||1|| ⋅ |*x*|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane**R**^{2}.### Composition algebras

Main article: Composition algebraEvery composition algebra

*A*has an involution*x*→*x** called its**conjugation**. The product in*A*of an element*x*and its conjugate*x** is written*N*(*x*) =*x x** and called the**norm of x**.The real numbers ℝ, complex numbers ℂ, and quaternions ℍ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element

*x*has a non-zero norm, then*x*has a multiplicative inverse given by*x**/*N*(*x*).## Notes

- ^
^{a}^{b}^{Every composition algebra A has an }involution*x*→*x** called its**conjugation**. The product in*A*of an element*x*and its conjugate*x** is written*N*(*x*) =*x x** and called the**norm of x**.The real numbers ℝ, complex numbers ℂ, and quaternions ℍ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element

*x*has a non-zero norm, then*x*has a multiplicative inverse given by*x**/*N*(*x*).

- $$

- 2.1 Real numbers

- 5.1 In mathematics, the