Sign function

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The sign function

sgn (x)

, also called the signum function, of a real number

x

is defined as
     

sgn (x)  :=  ⎰   ⎱     0  if x = 0, x | x |  if x ≠ 0.

This may also be written as, where [·] is the Iverson bracket,

     

sgn (x)  :=  [x ≥ 0] − [x ≤ 0]  =  ⎧   ⎨   ⎩     −1  if x < 0, 0  if x = 0, 1  if x > 0.

The signum of a nonzero real number

x

gives the real number which is closest to

x

on the unit 0-sphere of the real line, i.e. the real number with absolute value 1 which has the same sign as

x

.

Complex signum

The complex signum of a complex number

z = a + b i = r e i  θ = | z |  e i  arg (z)

, where

| z |

is the complex norm and

arg (z) = θ

is the argument of a nonzero complex number (the argument of 0 being undefined), would generalize to
     

sgn (z)  :=  ⎰   ⎱     0  if z = 0, z | z | = e i  arg (z) = e i  θ  if z ≠ 0.

The complex signum of a nonzero complex number

z

gives the complex number which is closest to

z

on the unit circle (the unit 1-sphere) of the complex plane, i.e. the complex number with complex norm 1 which has the same argument as

z

.

See also

• Absolute value (

  | x |

  )
• Heaviside step function
• {{sgn}} (mathematical function template)
• {{abs}} (mathematical function template)