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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 sign function is an odd
Even and odd functions

In mathematics, even functions and odd functions are function s which satisfy particular symmetry relations, with respect to taking additive inverses....
 mathematical function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 that extracts the 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....
 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...
. To avoid confusion with the sine
Siné

Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
 function, this function is often called the signum function (from signum, Latin for "sign").

In mathematical expressions the sign function is often represented as sgn.

signum function 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...
 x is defined as follows:

real number can be expressed as the product of its absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 and its sign function: From equation (1) it follows that whenever x is not equal to 0 we have

The signum function is 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 absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x.






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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 sign function is an odd
Even and odd functions

In mathematics, even functions and odd functions are function s which satisfy particular symmetry relations, with respect to taking additive inverses....
 mathematical function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 that extracts the 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....
 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...
. To avoid confusion with the sine
Siné

Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
 function, this function is often called the signum function (from signum, Latin for "sign").

In mathematical expressions the sign function is often represented as sgn.

Definition

The signum function 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...
 x is defined as follows:

Properties

Any real number can be expressed as the product of its absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 and its sign function: From equation (1) it follows that whenever x is not equal to 0 we have

The signum function is 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 absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x. The numbers cancel and all we are left with is the sign of x.

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory
Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
, the derivative of the signum function is two times the Dirac delta function
Dirac delta function

The Dirac delta or Dirac's delta is a mathematics construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function d that has the value 0 everywhere except at x = 0 where its value is infinity in such a way that its total integral is 1....
,

The signum function is related to 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....
 H1/2(x) thus: where the 1/2 subscript of the step function means that H1/2(0) = 1/2. The signum can also be written using the Iverson bracket
Iverson bracket

In mathematics, the Iverson bracket is a convenient notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise....
 notation: For , a smooth approximation of the sign function is See Heaviside step function – Analytic approximations
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....
.

Complex signum

The signum function can be generalized to complex numbers as for any z ? except z = 0. The signum of a given complex number z is the point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 on the unit circle
Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
 of 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....
 that is nearest to z. Then, for z ? 0, where arg is the complex argument function
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
. For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines sgn 0 = 0.

Another generalization of the sign function for real and complex expressions is csgn, which is defined as:

We then have (except for z = 0):

Generalized signum function

At real values of , it is possible to define a generalized function
Generalized function

In mathematics, generalized functions are objects generalizing the notion of function s. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges....
–version of the signum function, such that everywhere, including at the point (unlike , for which ). This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization is the loss of commutativity
Commutativity

In mathematics, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many binary operations throughout mathematics, and many Mathematical proof depend on it....
. In particular, the generalized signum anticommutes with the delta-function, in addition, cannot be evaluated at ; and the special name, is necessary to distinguish it from the function . ( is not defined, but .)

See also

  • Absolute value
    Absolute value

    In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
  • Heaviside function
  • Negative and non-negative numbers
    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....
  • Rectangular function
    Rectangular function

    The rectangular function is defined as:'Alternate definitions of the function define to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, :'...