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.

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## 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.

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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

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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.

**CITE THIS AS:**

**Stephanie Glen**. "Sign Function (Signum): Definition, Examples" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/sign-function/

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