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The
sign function , also called the
signum function, of a
real number is defined as

This may be compactly written as

where
is the
Iverson bracket.
"Sign" of complex number
The "sign" of a
complex number z = a + b i = r e^{i θ} =  z  e^{i arg(z)} 
, where
is the
absolute value of a complex number and
is the
argument of a nonzero complex number (the argument of 0 being undefined), would generalize to

The "sign" of a nonzero complex number would thus give the complex number on the unit circle of the
complex plane which has the same argument as
.
If we consider the real numbers as a subset of the complex numbers, then

See also