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.

.

## 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

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

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 also includes 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.

## Signum Function in Software

Every major Computer Algebra System (CAS) has an equivalent of a sign function. For example:

Examples:

- SIGN function in Derive,
- Signum and piecewise functions in Maple V ,
- Sign[x] in Mathematica.

Note that each of the major CAS uses a slightly different definition. This is because different branches of mathematics define the function in slightly different way (Jeffrey et. al, n.d.).

## References

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

Calculus I Homework: Calculating Limits Using the Limit Laws Page 1. Retrieved July 13, 2021 from: http://cda.morris.umn.edu/~mcquarrb/teachingarchive/M1101/HW/2.3.pdf

Jeffrey et al. Integration of the signum, piecewise and related functions. Retrieved July 13, 2021 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&rep=rep1&type=pdf

under special properties section, I believe you have the x and |x| flipped around. It’s x=sgn(x) * |x|

Thanks for catching that. It’s fixed :)

The UnitStep function in Mathematica evaluates to 1 for an input of, while that is not the case for the sign function defined in piecewise form. In Mathematica, the sign function is just Sign[x].

Thanks for the info!