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# 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 |
= ei  arg (z) = ei  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
.