Sign Function (Signum): Definition, Examples

Calculus Definitions >

The sign function (or signum function) is a special function which returns:

• 1 for all x > 0 and
• – 1 for all x < 0.

For x = 0, the value of the sign function is just zero.

It is a real-valued step function that tells us, numerically, whether a particular value of x is positive, negative, or zero.

The sign function.

Sign Function: Definition

For every real number x, the sign function sgn(x) is defined as:

Another definition of the signum function groups zero with the positive numbers. Under that definition,

• sgn(x) = 1 for x ≥ 0, and
• sgn(x) = -1.

Special Properties

The absolute value of any real number x can be written in terms of the signum function and the number itself.

.

This means we can also write the signum function as

Derivative of the Sign Function

For any x not equal to zero, the derivative of x is equal to the sign function. The derivative of the sign function is just equal to zero, except at zero, where the derivative does not exist.

Generalizing to Complex Numbers

The signum function doesn’t only work for real numbers; it can also be defined for complex numbers, but there it needs a broader definition. We define the signum function over the complex numbers (which, remember, include the reals) as:

If z ≠ 0, and if z is equal to zero, we say

.

A quick check should convince you that this definition is a reasonable generalization of what we’ve already defined over a smaller space.

There is another generalization that might be more intuitive to use, though it is not as much an extension of the signum function as an extension of the ideas behind the sign function. This is written csgn, and is defined as

where Im(z) is the imaginary part of a complex number z, and Re(z) the real part.

References

Signum Function. Retrieved from http://www.ai.mit.edu/projects/iiip/doc/CommonLISP/HyperSpec/Body/fun_signum.html on December 16, 2018.

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