Sign function

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The sign function or signum function (from Latin signum 'sign' ) is a function in mathematics that assigns its sign to a real or complex number .

Sign function on the real numbers


Sign function graph

The real sign function maps the set of real numbers into the set and is usually defined as follows:

She assigns the positive numbers the value +1, the negative numbers the value −1 and the 0 the value 0.

In computing technology applications, the special position of 0 is sometimes dispensed with; this is then assigned to the positive, negative or both number ranges. This allows the sign of a number to be encoded in a single bit . Since the zero is a zero set below the Lebesgue measure , this is not important for all practical applications.

In the event that is set, there is the following relationship to the Heaviside function :

Calculation rules

By differentiating between cases it is easy to prove:

  • For everyone with amount applies .
for everyone .
  • If is a constant and an odd function , then
  • For the transition to the reciprocal number is compatible with the signum function and does not change its value:
for everyone .
  • The signum function is compatible with multiplication:
for everyone .
for everyone .

From the last two calculation rules mentioned, it follows, for example, that the signum function in an argument composed of any number of factors can be replaced by a factor without changing the function value:

for any .

Derivative and integral

The sign function is not continuous at the 0 position.

The sign function is not continuous at this point and therefore not classically differentiable there. For all other places the sign function can be differentiated with . The sign function also has no weak derivative . However, it is differentiable in the sense of distributions , and its derivative is , where denotes the delta distribution .

Furthermore applies to everyone

The sign function is also the weak derivative of the absolute value function .

Sign function on the complex numbers


Signum of four complex numbers

Compared to the sign function of real numbers, the following extension to complex numbers is rarely considered:

The result of this function is on the unit circle and has the same argument as the output value, in particular applies

Example: (red in the picture)

Calculation rules

The following calculation rules apply to the complex sign function:

For all complex numbers and the following applies:

  • for all where denotes the amount of ;
  • where the slash denotes the complex conjugation ;
  • , in particular
    • for positive real ones ,
    • for negative reals ,
    • ;
  • .
  • If is, also applies


  • Königsberger: Analysis 1 . 6th edition. Springer, Berlin 2003, ISBN 3-540-40371-X , p. 101 .
  • Hildebrandt: Analysis 1 . 2nd Edition. Springer, Berlin 2005, ISBN 3-540-25368-8 , pp. 133 .

Web links

Wikibooks: Algorithm Collection: Number Theory: Signum  - Learning and Teaching Materials