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The

**sign function** , also called the

**signum function**, of a

real number is defined as

- ${\begin{array}{l} \operatorname {sgn}(x):={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}\end{array}}$

This may be compactly written as

- ${\begin{array}{l} \operatorname {sgn}(x):=[x\geq 0]-[x\leq 0],\quad x\in \mathbb {R} ,}\end{array}}$

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

- $\operatorname {sgn}(z):={\begin{cases}0&{\text{if }}z=0,\\e^{i\arg(z)}=e^{i\theta }&{\text{if }}z\neq 0.\end{cases}}$

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

- $\operatorname {sgn}(x):={\begin{cases}e^{i\pi }&{\text{if }}x<0,\\0&{\text{if }}x=0,\\e^{i0}&{\text{if }}x>0.\end{cases}}$

## See also