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In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function. The sign function is often represented as sgn and can be defined thus:

$\backslash sgn\; x\; =\; \backslash left\backslash \{\; \backslash begin\{matrix\}$

-1 & : & x < 0 \\ 0 & : & x = 0 \\ 1 & : & x > 0 \end{matrix} \right.

or using the Iverson bracket notation:

Any real number can be expressed as the product of its absolute value and its sign function:

$x\; =\; (\; \backslash sgn\; x\; )\; |x|.\; \backslash qquad\; \backslash qquad\; (1)$

From equation (1) it follows that whenever x is not equal to 0 we have

$\backslash sgn\; x\; =\; \{x\; \backslash over\; |x|\}\; \backslash qquad\; \backslash qquad\; (2)$

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero):

$\{d\; |x|\; \backslash over\; dx\}\; =\; \{x\; \backslash over\; |x|\}.$

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under a somewhat generalised notion of differentiation (see distribution), we can say that the derivative of the signum function is two times the Dirac delta function,

$\{d\; \backslash \; \backslash sgn\; x\; \backslash over\; dx\}\; =\; 2\; \backslash delta\; (x).$

The signum function is related to the Heaviside step function H_1/2(x) thus

$\backslash sgn\; x\; =\; 2\; H\_\{1/2\}(x)\; -\; 1,\; \backslash ,$

where the 1/2 subscript of the step function means that H_1/2(0) = 1/2

Complex Signum

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The signum function can be generalized to complex numbers as

$\backslash sgn\; z\; =\; \backslash frac\{z\}\{|z|\}$

for any z ∈ $\backslash mathbb\{C\}$ except z=0. The signum of a given complex number, z is the point on the unit circle of the complex plane that is nearest to z. For this reason, all numbers with the same signum form a line that passes through their signum and has a removable singularity at the origin. Another consequence of the fact that all complex numbers that represent the signum of another number lie on the unit circle of the complex plane is that for some n^th root of unity.

$\backslash sgn\; (r)\; =\; r$

Because zero has an equal distance to all points on the uinit circle, it is generally given the signum 0.

See also

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• Negative and non-negative numbers
• Absolute value

Elementary special functions

Funkce signum | Signum (Mathematik) | Función signo | Funzione segno | Szignumfüggvény | Signum (wiskunde) | Signum | Сигнум функција | Signumfunktionen | İşaret fonksiyonu | 符号函数