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The
sign function , also called the
signum function, of a
real number is defined as
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
gives the real number which is closest to
on the unit
0sphere of the
real line, i.e. the real number with absolute value
1 which has the same sign as
.
Complex signum
The complex signum of a
complex number z = a + b i = r e i θ =  z  e i arg (z) 
, where
is the
complex norm and
is the
argument of a nonzero complex number (the argument of
0 being undefined), would generalize to
sgn (z) :=

0  if z = 0, 
= e i arg (z) = e i θ  if z ≠ 0.



The complex signum of a nonzero complex number
gives the complex number which is closest to
on the unit circle (the unit
1sphere) of the
complex plane, i.e. the complex number with complex norm
1 which has the same argument as
.
See also