Absolute value

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Since the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in equation (1) above:
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 right triangle...

 that the absolute value of a complex number is the distance in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established 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. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

, 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 (2)–(10) above. In addition, If


and


is the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

 of z, then it is easily seen that


and


with the last formula being the complex analogue of equation (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 ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...

 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 \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...

 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". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 everywhere. It is differentiable

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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 everywhere except for x = 0. It is monotonically decreasing

Monotonic function

In mathematics, a monotonic function is a function that 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 (−∞, 0] and monotonically increasing on the interval [0, ∞). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible.

Both the real and complex functions are idempotent.

It is a nonlinear function.

It is also a convex

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

 function.

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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of the real absolute value function is given by


for x ≠ 0 (cf. signum function). 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.

The complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 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 Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...

.

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.

Antiderivative

The absolute value function is also integrable. Its antiderivative

Antiderivative

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

 is


as evidenced by the following (using integration by parts and the fact that x² = |x|²):

Relationship to other functions

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)|x|. The real absolute value function is also related to a form of the Heaviside step function

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one 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,

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, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

 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, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

 between two points


and


in Euclidean n-space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 is defined as:

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

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 R with a total order ≤ such that for all a, b, and c in R:* if a ≤ b then a + c ≤ b + c.* if 0 ≤ a and 0 ≤ b then 0 ≤ ab....

. 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 a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

 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...

.

Fields

The fundamental properties of the absolute value for real numbers given in (2)–(5) 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 a commutative ring whose nonzero elements form a group under multiplication. As such it 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 | x | from D to the real numbers R satisfying:* | x | ≥ 0,*...

(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 for all x, y, z in F.

• 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.

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

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V over a field F, represented as ||V||, 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 the triangle inequality

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

In the case of Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 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, 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...

, and is the p-norm 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 an additional structure called an 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 or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 is identified with the Euclidean plane R².

Smooth approximation

The absolute value function does not have a derivative at x=0. Sometimes, a smooth approximation is required that has to be smooth at x=0. One of such approximations is given by:
where k>0. The approximation improves as k increases. k=10 is good for many practical purposes.