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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the absolute value or modulus of a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
$x$, is the non-negative value without regard to its sign. Namely, $, x, =x$ if is a positive number, and $, x, =-x$ if $x$ is negative (in which case negating $x$ makes $-x$ positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s, ordered rings, fields and
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. The absolute value is closely related to the notions of magnitude,
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
, and norm in various mathematical and physical contexts.

# Terminology and notation

In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,
Oxford English Dictionary The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a c ...
, Draft Revision, June 2008
and it was borrowed into English in 1866 as the Latin equivalent ''modulus''. The term ''absolute value'' has been used in this sense from at least 1806 in French and 1857 in English. The notation , with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for ''absolute value'' include ''numerical value'' and ''magnitude''. In programming languages and computational software packages, the absolute value of ''x'' is generally represented by abs(''x''), or a similar expression. 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, it denotes its
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
. 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 or sup norm of a vector although double vertical bars with subscripts respectively) are a more common and less ambiguous notation.

# Definition and properties

## Real numbers

For any the absolute value or modulus is denoted , with a vertical bar on each side of the quantity, and is defined as $, x, = \begin x, & \text x \geq 0 \\ -x, & \text x < 0. \end$ The absolute value is thus always either a positive number or
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
, but never negative. When $x$ itself is negative then its absolute value is necessarily positive From an
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
point of view, the absolute value of a real number is that number's
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from zero along the real number line, and more generally the absolute value of the difference of two real numbers (their absolute difference) is the distance between them. 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 $, x, = \sqrt.$ This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers. 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: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that $, a+b, =s\left(a+b\right)$ with its sign chosen to make the result positive. Now, since $-1 \cdot x \le , x,$ it follows that, whichever of $\pm1$ is the value one has $s \cdot x\leq , x,$ for all Consequently, $, a+b, =s \cdot \left(a+b\right) = s \cdot a + s \cdot b \leq , a, + , b,$, as desired. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above. Two other useful properties concerning inequalities are: These relations may be used to solve inequalities involving absolute values. For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
on the real numbers.

## Complex numbers

Since the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. This can be computed using the Pythagorean theorem: for any complex number $z = x + iy,$ where $x$ and $y$ are real numbers, the absolute value or modulus is and is defined by $, z, = \sqrt=\sqrt,$ the
Pythagorean addition In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides a and b, this leng ...
of $x$ and $y$, where $\operatorname\left(z\right)=x$ and $\operatorname\left(z\right)=y$ denote the real and imaginary parts respectively. When the is zero, this coincides with the definition of the absolute value of the When a complex number $z$ is expressed in its polar form its absolute value Since the product of any complex number $z$ and its with the same absolute value, is always the non-negative real number the absolute value of a complex number $z$ is the square root which is therefore called the absolute square or ''squared modulus'' $, z, = \sqrt.$ This generalizes the alternative definition for reals: The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity $, z, ^2 = , z^2,$ is a special case of multiplicativity that is often useful by itself.

# Absolute value function

The real absolute value function is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
everywhere. It is differentiable everywhere except for . It is monotonically decreasing on the interval and monotonically increasing on the interval . Since a real number and its opposite have the same absolute value, it is an
even function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
, and is hence not invertible. The real absolute value function is a piecewise linear, convex function. For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself).

## Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions: :$, x, = x \sgn\left(x\right),$ or :$, x, \sgn\left(x\right) = x,$ and for , :$\sgn\left(x\right) = \frac = \frac.$

## Derivative

The real absolute value function has a derivative for every , but is not differentiable at . Its derivative for is given by the step function:Bartle and Sherbert, p. 163 :$\frac = \frac = \begin -1 & x<0 \\ 1 & x>0. \end$ The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of  at  is the interval . The complex absolute value function is continuous everywhere but complex differentiable ''nowhere'' because it violates the Cauchy–Riemann equations. The second derivative of  with respect to  is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
.

## Antiderivative

The
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
(indefinite integral) of the real absolute value function is :$\int \left, x\ dx = \frac + C,$ where is an arbitrary
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable ( holomorphic) functions, which the complex absolute value function is not.

# Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
between two points :$a = \left(a_1, a_2, \dots , a_n\right)$ and :$b = \left(b_1, b_2, \dots , b_n\right)$ in Euclidean -space is defined as: :$\sqrt.$ This can be seen as a generalisation, since for $a_1$ and $b_1$ real, i.e. in a 1-space, according to the alternative definition of the absolute value, :$, a_1 - b_1, = \sqrt = \sqrt,$ and for $a = a_1 + i a_2$ and $b = b_1 + i b_2$ complex numbers, i.e. in a 2-space, : The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function on a set is called a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
(or a ''distance function'') on , if it satisfies the following four axioms: :

# Generalizations

## Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if  is an element of an ordered ring ''R'', then the absolute value of , denoted by , is defined to be: :$, a, = \left\\left\{ \begin\left\{array\right\}\left\{rl\right\} a, & \text\left\{if \right\} a \geq 0 \\ -a, & \text\left\{if \right\} a < 0. \end\left\{array\right\}\right.$ where is the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
of , 0 is the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
, and < and ≥ have the usual meaning with respect to the ordering in the ring.

## Fields

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. A real-valued function  on a field  is called an ''absolute value'' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation'') if it satisfies the following four axioms: :{, cellpadding=10 , - , $v\left(a\right) \ge 0$ , Non-negativity , - , $v\left(a\right) = 0 \iff a = \mathbf\left\{0\right\}$ , Positive-definiteness , - , $v\left(ab\right) = v\left(a\right) v\left(b\right)$ , Multiplicativity , - , $v\left(a+b\right) \le v\left(a\right) + v\left(b\right)$ , Subadditivity or the triangle inequality Where 0 denotes the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
of . It follows from positive-definiteness and multiplicativity that , where 1 denotes the
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of . The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If is an absolute value on , then the function  on , defined by , is a metric and the following are equivalent: * satisfies the
ultrametric In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
inequality $d\left(x, y\right) \leq \max\left(d\left(x,z\right),d\left(y,z\right)\right)$ for all , , in . * $\left\{ v\left( \sum_{k=1}^n \mathbf{1}\right) : n \in \N \right\}$ is bounded in R. * $v\left\left(\left\{\textstyle \sum_\left\{k=1\right\}^n \right\} \mathbf\left\{1\right\}\right\right) \le 1\$ for every $n \in \N$. * $v\left(a\right) \le 1 \Rightarrow v\left(1+a\right) \le 1\$ for all $a \in F$. * $v\left(a + b\right) \le \max \\left\{v\left(a\right), v\left(b\right)\\right\}\$ for all $a, b \in F$. An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.Shechter
pp. 260–261

## Vector spaces

Again 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field , represented as , is called an absolute value, but more usually a norm, if it satisfies the following axioms: For all  in , and , in , :{, cellpadding=10 , - , $\, \mathbf\left\{v\right\}\, \ge 0$ , Non-negativity , - , $\, \mathbf\left\{v\right\}\, = 0 \iff \mathbf\left\{v\right\} = 0$ , Positive-definiteness , - , $\, a \mathbf\left\{v\right\}\, = \left, a\ \left\, \mathbf\left\{v\right\}\right\,$ , Positive homogeneity or positive scalability , - , $\, \mathbf\left\{v\right\} + \mathbf\left\{u\right\}\, \le \, \mathbf\left\{v\right\}\, + \, \mathbf\left\{u\right\}\,$ , Subadditivity or the triangle inequality The norm of a vector is also called its ''length'' or ''magnitude''. In the case of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
$\mathbb\left\{R\right\}^n$, the function defined by :$\, \left(x_1, x_2, \dots , x_n\right) \, = \sqrt\left\{\textstyle\sum_\left\{i=1\right\}^\left\{n\right\} x_i^2\right\}$ is a norm called the Euclidean norm. When the real numbers $\mathbb\left\{R\right\}$ are considered as the one-dimensional vector space $\mathbb\left\{R\right\}^1$, the absolute value is a norm, and is the -norm (see Lp space) for any . In fact the absolute value is the "only" norm on $\mathbb\left\{R\right\}^1$, in the sense that, for every norm on $\mathbb\left\{R\right\}^1$, . The complex absolute value is a special case of the norm in an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, which is identical to the Euclidean norm when the complex plane is identified as the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
$\mathbb\left\{R\right\}^2$.

## Composition algebras

Every composition algebra ''A'' has an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
''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 $\mathbb\left\{R\right\}$, complex numbers $\mathbb\left\{C\right\}$, and quaternions $\mathbb\left\{H\right\}$ are all composition algebras with norms given by
definite quadratic form In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical de ...
s. The absolute value in these
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
s is given by the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of the composition algebra norm. In general the norm of a composition algebra may be a
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
that is not definite and has
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...
s. However, as in the case of division algebras, when an element ''x'' has a non-zero norm, then ''x'' has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
given by ''x''*/''N''(''x'').

* Least absolute values

# References

* Bartle; Sherbert; ''Introduction to real analysis'' (4th ed.), John Wiley & Sons, 2011 . * Nahin, Paul J.; ''An Imaginary Tale''; Princeton University Press; (hardcover, 1998). . * Mac Lane, Saunders, Garrett Birkhoff, ''Algebra'', American Mathematical Soc., 1999. . * Mendelson, Elliott, ''Schaum's Outline of Beginning Calculus'', McGraw-Hill Professional, 2008. . * O'Connor, J.J. and Robertson, E.F.
"Jean Robert Argand"
* Schechter, Eric; ''Handbook of Analysis and Its Foundations'', pp. 259–263
"Absolute Values"