The sign of a real number, also called sgn or signum, is 
for a negative number (i.e.,
one with a minus sign "
"), 0 for the number zero, or
for a positive
number (i.e., one with a plus sign "
"). In other words, for real 
,
 |
(1)
|
For real 
, this can be written
 |
(2)
|
and satisfies
 |
(3)
|
for real 
can also be defined as
 |
(4)
|
where 
is the Heaviside
step function.
The sign function is implemented in the Wolfram Language for real 
as Sign[x].
For nonzero complex numbers, Sign[z]
returns 
, where 
is the complex modulus
of 
.
For a nonzero complex number 
, the point 
lies on the unit
circle and on the ray from the origin
through 
. Thus it retains the complex
argument of 
and has complex modulus 1.
can also be interpreted as an unspecified
point on the unit circle in the complex plane (Rich and Jeffrey 1996).
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