Sign function

Signum function

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:

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

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

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

From equation (1) it follows that

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

but equation (2) is indeterminate when x is set to zero.

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

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

Also, the derivative of the signum function is two times the Dirac delta function,

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

The signum function is related to the Heaviside step function h_0.5(x) thus

sgnx = 2h_0.5(x) - 1,

where the 0.5 subscript of the step function means that h_0.5(0) = 0.5.

Also, if the step function h₀(x) is thought of as a mathematical switch, with h₀(x) = 0, then the signum function can be expressed as

$\sgn x = | h_0 (x) | (-1)^{h_0(-x)}.$

Sign function

See also