Absolute value

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Absolute value

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the absolute value (or modulus) of a real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
is its numerical value without regard to its sign

Negative and non-negative numbers

A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
. So, for example, 3 is the absolute value of both 3 and -3.

The absolute value of a number is denoted by .

Generalizations 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

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, the quaternion

Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
s, ordered ring

Ordered ring

In abstract algebra, an ordered ring is a commutative ring with a total order = such that for all a, b, and c in R:* if a = b then a + c = b + c....
s, field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
s and vector space

Vector space

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Terminology and notation

Jean-Robert Argand

Jean-Robert Argand was a non-professional mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Complex plane....
introduced the term "module" 'unit of measure' in French in 1806 specifically for the complex absolute value and it was borrowed into English in 1866 as the Latin equivalent "modulus".

The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English.

The notation | a | was introduced by Karl Weierstrass

Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a Germany mathematics who is often cited as the "father of modern mathematical analysis"....
in 1841.

Other names for absolute value include "the numerical value" and "the magnitude", that is, ignoring the sign.

Real numbers

For any real number

Real number

Vertical bar

The vertical bar has various names including the pipe , verti-bar, vbar, stick, vertical line, vertical slash, think colon, or divider line by others....
on each side of the quantity) and is defined as

As can be seen from the above definition, the absolute value of a is always either positive or zero

0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
, but never negative

Negative and non-negative numbers

A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
. The same notation is used with sets to denote cardinality

Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
; the meaning depends on context.

From an analytic geometry

Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
point of view, the absolute value of a real number is that number's distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
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 generalization of the absolute value of the difference (see "Distance" below).

Since the square-root function is normally defined as the positive square root, , which is sometimes even used as a definition of absolute value.

The absolute value has the following four fundamental properties:

	Non-negativity
	Positive-definiteness
	Multiplicativeness
	Subadditivity

Other important properties of the absolute value include:

	Symmetry Symmetry Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
	Identity of indiscernibles Identity of indiscernibles The identity of indiscernibles is an ontology principle which states that two or more object s or entity are identical , if they have all their property in common.... (equivalent to positive-definiteness)
	Triangle inequality Triangle inequality In mathematics, the triangle inequality states that for any triangle, the length of a given side must be less than the sum of the other two sides but greater than the difference between the two sides.... (equivalent to subadditivity)
	Preservation of division (equivalent to multiplicativeness)
	(equivalent to subadditivity)

If b > 0, two other useful inequalities are:

These relations may be used to solve inequalities involving absolute values:

Complex numbers

Since the complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s are not ordered

Ordered set

Ordered set is used with distinct meanings in order theory.*A Set with a binary relation R on its elements that is reflexive relation , antisymmetric relation and transitive relation is described as a partially ordered set or poset....
, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in Proposition 1: can be seen as motivating the following definition.

For any complex number

where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is defined as

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since:

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
that the absolute value of a complex number is the distance in the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
of that complex number from the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

The complex absolute value shares all the properties of the real absolute value given in Propositions 2 and 3 above. In addition, If

and

is the complex conjugate

Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
of z, then it is easily seen that

and

with the last formula being the complex analogue of Proposition 1 mentioned above in the real case.

The absolute square of z is defined as

Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism

Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ?: V ? V, and an endomorphism of a group G is a group homomorphism ?: G ? G....
of the multiplicative group

Multiplicative group

In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group whose binary operation is written in multiplicative notation ,...
of the complex numbers.

Absolute value functions

The real absolute value function is continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
everywhere. It is differentiable

Derivative

Monotonic function

In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
on the interval (−8, 0] and monotonically increasing on the interval [0, 8). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible.

The complex

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
absolute value function is continuous everywhere but (complex) differentiable nowhere; it violates the Cauchy-Riemann equations

Cauchy-Riemann equations

In mathematics, the Cauchy?Riemann differential equations in complex analysis, named after Augustin Louis Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a Necessary and sufficient conditions condition for a differentiable function to be holomorphic function in an open set....
.

Both the real and complex functions are idempotent.

It is a nonlinear function.

Ordered rings

The definition of absolute value given for real numbers above can easily be extended to any ordered ring

Ordered ring

In abstract algebra, an ordered ring is a commutative ring with a total order = such that for all a, b, and c in R:* if a = b then a + c = b + c....
. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by | a |, is defined to be:

where −a is the additive inverse

Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
of a, and 0 is the additive identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
.

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

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
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

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem....
between two points

and

in Euclidean n-space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
is defined as:

This can be seen to be a generalization of | a − b |, since if a and b are real, then by Proposition 1,

while if

and

are complex numbers, then

The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the 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 in Propositions 2 and 3 above, can be seen to motivate the more general notion of a distance function as follows:

A real valued function d on a set X × X is called a distance function (or a metric) on X, if it satisfies the following four axioms:

	Non-negativity
	Identity of indiscernibles
	Symmetry
	Triangle inequality

Derivatives

The derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
of the real absolute value function is the signum function, sgn(x), which is defined as

for x ? 0. The absolute value function is not differentiable at x = 0. For applications in which a well-defined derivative may be needed, however, the subderivative

Subderivative

In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in the study of convex functions, often in connection to convex optimization....
is well-defined at zero. Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. Therefore x = sgn(x)abs(x). The signum function is a form of the Heaviside step function

Heaviside step function

The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
used in signal processing, defined as:

where the value of the Heaviside function at zero is conventional. So for all nonzero points on the real number line,

The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |x| with respect to x is zero everywhere except zero, where it is undefined.

The absolute value function is also integrable. Its antiderivative

Antiderivative

In calculus, an antiderivative, primitive or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f....
is

Proof

Fields

The fundamental properties of the absolute value for real numbers given in Proposition 2 above, can be used to generalize the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
F is called an absolute value
Absolute value (algebra)

In mathematics, an absolute value is a function which measures the "size" of elements in a Field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping | ⋅ | from D to the real numbers R satisfying:...
(also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms:

	Non-negativity
	Positive-definiteness
	Multiplicativeness
	Subadditivity or the triangle inequality

Where 0 denotes the additive identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
of F. It follows from positive-definiteness and multiplicativeness that v(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the following are equivalent:

d satisfies the ultrametric inequality d(x, y) < max.

is bounded
Bounded set

In mathematical analysis and related areas of mathematics, a Set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded....
in R.

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 field

In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy....
.

Vector spaces

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalize the notion to an arbitrary vector space.

A real valued function || · || on a vector space

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
V over a field F, is called an absolute value (or more usually a norm) if it satisfies the following axioms:

For all a in F, and v, u in V,

	Non-negativity
	Positive-definiteness
	Positive homogeneity or positive scalability
	Subadditivity or triangle inequality

The norm of a vector is also called its length or magnitude.

In the case of Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
Rⁿ, the function defined by

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R¹

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, the absolute value is a norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
for any p. In fact the absolute value is the "only" norm on R¹, in the sense that, for every norm || · || on R¹, || x || = || 1 || · | x |. The complex absolute value is a special case of the norm in an inner product space

Inner product space

In mathematics, an inner product space is a vector space with the additional Mathematical structure of inner product. This additional structure associates each pair of vectors in the space with a Scalar quantity known as the inner product of the vectors....
. It is identical to the Euclidean norm, if the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
is identified with the Euclidean plane R².

Algorithms

In the C programming language

C (programming language)

C is a general-purpose computer programming language originally developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories to implement the Unix operating system....
, the abs, labs, llabs (in C99), fabs, fabsf, and fabsl functions, found in math.h

Math.h

math.h is a header file in the C standard library of C programming language designed for basic mathematical operations. Most of the functions involve the use of floating point numbers....
, compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input:

int abs (int i)

The floating-point versions are trickier, as they have to contend with special codes for infinities and not-a-number.

Using assembly language

Assembly language

An assembly language is a low-level language for programming computers. It implements a symbolic representation of the numeric machine codes and other constants needed to program a particular CPU architecture....
, it is possible to take the absolute value of a register

Processor register

In computer architecture, a processor register is a small amount of Computer storage available on the CPU whose contents can be accessed more quickly than storage available elsewhere....
in just three instructions (example shown for a 32-bit register on an x86 architecture

X86 architecture

The generic term x86 refers to the most commercially successful instruction set architecture in the history of personal computing. It derived from the model numbers, ending in "86", of the first few processor generations Backward compatibility with the original Intel 8086....
, Intel syntax):

cdq xor eax, edx sub eax, edx

cdq extends the sign bit of eax into edx. If eax is nonnegative, then edx becomes zero, and the latter two instructions have no effect, leaving eax unchanged. If eax is negative, then edx becomes 0xFFFFFFFF, or -1. The next two instructions then become a two's complement

Two's complement

The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two .A two's-complement system or two's-complement arithmetic is a system in which negative numbers are represented by the two's complement of the absolute value; this system is the most common Signed number r...
inversion, giving the absolute value of the negative value in eax.

Absolute value

Terminology and notation

Real numbers

Complex numbers

Absolute value functions

Ordered rings

Distance

Derivatives

Fields

Vector spaces

Algorithms

See also

External links