Evaluate numerically:

Complex number inputs:

Evaluate to high precision:

For real inputs, the result is exact:

For complex inputs, the precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Sign function using MatrixFunction:

Sign can be used with Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Values of Sign at fixed points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Find a value of for which the :

Visualize the result:

Plot on the real axis:

Plot the real and imaginary parts of the function:

Visualize Sign in three dimensions:

Plot the real part of the function:

Plot the imaginary part of the function:

Sign is defined for all real and complex inputs:

Function range of Sign for real inputs:

The range over the complex plane is the unit circle plus the origin:

Sign is an odd function:

Sign has mirror symmetry :

Sign is not a differentiable function:

The difference quotient does not have a limit in the complex plane:

There is only a limit in certain directions, for example, the real direction:

Use RealSign to obtain this real-differentiable result:

Sign is not an analytic function:

It has both singularities and discontinuities:

Over the complex plane, it is singular everywhere but still discontinuous only at the origin:

Sign is nonincreasing:

Sign is not injective:

Sign is not surjective:

Sign is neither non-negative nor non-positive:

Sign is neither convex nor concave:

TraditionalForm formatting:

Expand assuming real variables x and y:

Simplify Sign using appropriate assumptions:

Express a complex number as a product of Sign and Abs:

is equal to :

for all non-zero :