TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the absolute value or modulus of a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, denoted , is the
non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...
value of  without regard to its
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
. Namely, if is
positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign, the sign "+" used to indicate a positive number * Positive (electricity), a po ...
, and if is negative (in which case is positive), and . 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 Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, the
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s,
ordered ring 350px, The real numbers are an ordered ring which is also an ordered field. The integers">ordered_field.html" ;"title="real numbers are an ordered ring which is also an ordered field">real numbers are an ordered ring which is also an ordered field. ...
s,
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
and
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. The absolute value is closely related to the notions of
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
,
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

, and
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
in various mathematical and physical contexts.

# Terminology and notation

In 1806,
Jean-Robert ArgandJean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore Image:Libraria Carturesti Carusel - Interior ziua.jpg, 250px, Cărturești Carusel, a bookshop in a historical building ...
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 principal historical dictionary A historical dictionary or dictionary on historical principles is a dictionary which deals not only with the latterday meanings of words but also the historica ...
, 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 The vertical bar, , is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in mathematical logic, logic), pipe, vbar, stick, vertical line, bar, verti-bar ...
on each side, was introduced by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; when applied to a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
, it denotes its
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. 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 algebraIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
or
sup norm Image:Vector norm sup.svg, frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real number, real- or complex number, ...
of a vector in $\R^n$, although double vertical bars with subscripts ($\, \cdot\, _2$ and $\, \cdot\, _\infty$, respectively) are a more common and less ambiguous notation.

# Definition and properties

## Real numbers

For any
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, the absolute value or modulus of  is denoted by (a
vertical bar The vertical bar, , is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in mathematical logic, logic), pipe, vbar, stick, vertical line, bar, verti-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 of  is thus always either
positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign, the sign "+" used to indicate a positive number * Positive (electricity), a po ...
or
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, but never negative: when 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
point of view, the absolute value of a real number is that number's
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

from zero along the
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil i ... , and more generally the absolute value of the difference of two real numbers is the distance between them. The notion of an abstract distance function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol In mathematics, the radical sign, radical symbol, root symbol, radix, or surd is a Mathematical notation, symbol for the square root or Nth root, higher-order root of a number. The square root of a number x is written as :\sqrt, while the nth roo ... represents the unique ''positive'' square root (when applied to a positive number), it follows that :$, x, = \sqrt$ 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 one of the two alternatives of taking as either or guarantees that $s \cdot \left(a+b\right) = , a+b, \geq 0.$ Now, since $-1 \cdot x \le , x,$ and $+1 \cdot x \le , x,$, it follows that, whichever is the value of , one has $s \cdot x\leq , x,$ for all real $x$. Consequently, $, a+b, =s \cdot \left(a+b\right) = s \cdot a + s \cdot b \leq , a, + , b,$, as desired. (For a generalization of this argument to complex numbers, see "Proof of the triangle inequality for complex numbers" below.) 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: :$, a, \le b \iff -b \le a \le b$ :$, a, \ge b \iff a \le -b\$ or $a \ge b$ 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 The absolute difference of two real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ... between arbitrary real numbers, the standard metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ... on the real numbers. ## Complex numbers Since the complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ... : for any complex number :$z = x + iy,$ where and are real numbers, the absolute value or modulus of is denoted and is defined by :$, z, = \sqrt=\sqrt,$ where Re(''z'') = ''x'' and Im(''z'') = ''y'' denote the real and imaginary parts of ''z'', respectively. When the imaginary part is zero, this coincides with the definition of the absolute value of the real number . When a complex number is expressed in its polar form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... as :$z = r e^,$ with $r = \sqrt \ge 0$ (and is the argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ... (or phase) of ''z''), its absolute value is :$, z, = r .$ Since the product of any complex number and its complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ... with the same absolute value, is always the non-negative real number $\left\left(x^2 + y^2\right\right)$, the absolute value of a complex number is the square root of $z \cdot \overline,$ which is therefore called the absolute square . Each block represents one unit, , and the entire square represents , or the area of the square. In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squa ... or ''squared modulus'' of : :$, z, = \sqrt.$ This generalizes the alternative definition for reals: The complex absolute value shares the four fundamental properties given above for the real absolute value. In the language of group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ... , the multiplicative property may be rephrased as follows: the absolute value is a group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... from the multiplicative group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... of the complex numbers onto the group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... under multiplication of positive real numbersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ... . Importantly, the property of subadditivityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ... (" triangle inequality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... ") extends to any finite collection of complex This inequality also applies to infinite families In human society, family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of families is to maintain the w ... , provided that the infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... $\sum_^\infty z_k$ is absolutely convergent In mathematics, an Series (mathematics), infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a Real number, real or Complex number, ... . If Lebesgue integration In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... is viewed as the continuous analog of summation, then this inequality is analogously obeyed by complex-valued, measurable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... s $f: \R \to \C$ when integrated over a measurable subset (This includes Riemann-integrable functions over a bounded interval ### Proof of the complex triangle inequality The triangle inequality, as given by (), can be demonstrated by applying three easily verified properties of the complex numbers: Namely, for every complex number $z \in \Complex$, # there exists $c \in \Complex$ such that $, c, = 1$ and $, z, = c \cdot z$; # $\operatorname\left(z\right)\leq , z,$. Also, for a family of complex numbers In particular, # • if $\sum_k w_k \in \R$, then $\sum_k w_k =\sum_k \operatorname (w_k)$. • ''Proof of'' ()'':'' Choose $c \in \C$ such that $, c, = 1$ and $\left, \sum_k z_k\ = c \left(\sum_k z_k\right)$ (summed over The following computation then affords the desired inequality: :$\left, \sum_k z_k\\; \overset \; c\left\left(\sum_k z_k\right\right) = \sum_k cz_k\; \overset \;\sum_k\operatorname\left(cz_k\right)\; \overset \; \sum_k , cz_k, = \sum_k \left, c\ \left, z_k\ = \sum_k \left, z_k \ .$ It is clear from this proof that equality holds in () exactly if all the $c z_k$ are non-negative real numbers, which in turn occurs exactly if all nonzero $z_k$ have the same argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ... , i.e., $z_k = a_k\zeta$ for a complex constant $\zeta$ and real constants $a_k \geq 0$ for Since $f$ measurable implies that $, f,$ is also measurable, the proof of the inequality () proceeds via the same technique, by replacing $\sum_k(\cdot)$ with $\int_E (\cdot)\, dx$ and $z_k$ with # 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 ga ... everywhere. It is differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... 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 The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4. In mathematics, even functions and odd functions are function (mathematics), functions which satisfy par ... , and is hence not invertible. The real absolute value function is a , convex function (in green) is a convex set File:Convex polygon illustration2.svg, Illustration of a non-convex set. Illustrated by the above line segment whereby it changes from the black color to the red color. Exemplifying why this above set, illustrated in gr ... . Both the real and complex functions are idempotent Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ... . ## 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 In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... at . Its derivative for is given by the step function In mathematics, a function on the real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican ...

:Weisstein, Eric W. ''Absolute Value.'' From MathWorld – A Wolfram Web Resource.
/ref>Bartel 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 In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily Differentiable function, differentiable. Subderivatives arise in convex analysis, the study of convex functi ...
of  at  is the interval . The
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

absolute value function is continuous everywhere but
complex differentiable A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
''nowhere'' because it violates the
Cauchy–Riemann equations In the field of complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis ...
. The second derivative of  with respect to  is zero everywhere except zero, where it does not exist. As a
generalised functionIn mathematics, generalized functions are objects extending the notion of function (mathematics), functions. There is more than one recognized theory, for example the theory of distribution (mathematics), distributions. Generalized functions are esp ...
, the second derivative may be taken as two times the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
.

## Antiderivative

The
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
(indefinite integral) of the real absolute value function is :$\int \left, x\ dx = \frac + C,$ where is an arbitrary
constant of integration In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. Th ...

. This is not a
complex antiderivative In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex number, complex-valued function (mathematics), function ''g'' is a function whose complex derivative is ''g''. More precisely, given an open set U in the c ...
because complex antiderivatives can only exist for complex-differentiable (
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
) 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 measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

as follows: A real valued function on a set is called a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
(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 350px, The real numbers are an ordered ring which is also an ordered field. The integers">ordered_field.html" ;"title="real numbers are an ordered ring which is also an ordered field">real numbers are an ordered ring which is also an ordered field. ...
. 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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
of , 0 is the
additive identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of . It follows from positive-definiteness and multiplicativity that , where 1 denotes the
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Archimedean means of or pertaining to or named in honor of the Greece, Greek mathematics, mathematician Archimedes and may refer to: In mathematics: *Absolute value (algebra), Archimedean absolute value *Archimedean circle *Archimedean constant *Ar ...
.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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 , - , $\, a \mathbf\left\{v\right\}\, = , a, \, \mathbf\left\{v\right\}\,$ , 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
$\mathbb\left\{R\right\}^n$, 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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
. 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 Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, and is the -norm (see ) 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, which is identical to the Euclidean norm when the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
is identified as the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
$\mathbb\left\{R\right\}^2$.

## 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 input ...
''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 $\mathbb\left\{R\right\}$ are all composition algebras with norms given by
definite quadratic formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s. The absolute value in these
division algebra Division or divider may refer to: Mathematics *Division (mathematics) Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multi ...
s is given by the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the composition algebra norm. In general the norm of a composition algebra may be a
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
that is not definite and has
null vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. However, as in the case of division algebras, when an element ''x'' has a non-zero norm, then ''x'' has a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

given by ''x''*/''N''(''x'').

# 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"