Evaluate numerically:

Complex number input:

Evaluate to high precision:

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 Abs function using MatrixFunction :

Abs can be used with Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Values of Abs at fixed points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Exact inputs:

Find real values of for which TemplateBox[{x}, Abs]=2:

Substitute in the value of to create pairs:

Visualize the result:

Plot TemplateBox[{{1, +, , x}}, Abs] on the real axis:

Plot TemplateBox[{{1, +, {ⅈ, , x}}}, Abs] on the real axis:

Plot Abs in the complex plane:

Visualize Abs in three dimensions:

Use Abs to specify regions of the complex plane:

Abs is defined for all real and complex inputs:

The range of Abs is the non-negative reals:

This is true even in the complex plane:

Abs is an even function:

Abs 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:

This result, restricted to real inputs, is the derivative of RealAbs :

Abs is not an analytic function:

It has singularities but no discontinuities:

Over the complex plane, it is singular everywhere but still continuous:

Abs is neither nondecreasing nor nonincreasing:

Abs is not injective:

Abs is not surjective:

Abs is non-negative:

Abs is convex:

TraditionalForm formatting:

Expand assuming real variables x and y:

Simplify Abs using appropriate assumptions:

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

Express in terms of real and imaginary parts:

Abs commutes with real exponentiation:

This result is applied automatically for concrete powers:

Find the absolute value of a Root expression: