signum

Signum, also called the sign function, is a mathematical function that extracts the sign of a real number. It maps a real input to -1, 0, or 1, depending on whether the input is negative, zero, or positive.

For real numbers, sgn(x) = -1 if x < 0, sgn(x) = 0 if x = 0, and sgn(x) = 1 if x > 0. Equivalently, sgn(x) = x/|x| for x ≠ 0, with sgn(0) defined as 0 by convention. The sign function can be extended to complex numbers by sgn(z) = z/|z| for z ≠ 0, and sgn(0) = 0; in the complex case the result has magnitude 1 for nonzero inputs.

Properties include that the sign function is odd: sgn(-x) = -sgn(x), and it is discontinuous at 0, being piecewise constant elsewhere. It relates to the absolute value via |x| = sgn(x) · x. It can also be expressed in terms of the Heaviside step function H by sgn(x) = 2H(x) − 1 for x ≠ 0, with a chosen convention at x = 0.

In calculus, the derivative of sgn(x) is 0 for x ≠ 0 and is not defined at x = 0; in the theory of distributions, the derivative of sgn is 2δ, where δ is the Dirac delta distribution. Applications of the signum function appear in piecewise definitions, numerical algorithms, control theory, signal processing, and as a simple non-smooth tool in analysis.

Origin and usage: the term signum comes from Latin signum, meaning sign or mark. In programming and software, signum routines return the sign of a number, and in mathematics it serves as a foundational non-smooth function for building more complex constructs.