Abs
Abs[z]
gives the absolute value of the real or complex number z.
Examples
open allclose allBasic Examples (4)
Scope (28)
Numerical Evaluation (6)
Specific Values (6)
Values of Abs at fixed points:
Find real values of for which
:
Visualization (5)
Function Properties (5)
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:
TraditionalForm formatting:
Function Identities and Simplifications (6)
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:
Properties & Relations (16)
Abs is idempotent:
Abs is defined for all complex numbers:
RealAbs is defined only for real numbers:
Simplify expressions containing Abs:
Simplification of some identities involving Abs may require explicit assumptions that variables are real:
The assumptions may not be needed if RealAbs is used instead:
Abs is not a differentiable function:
RealAbs is differentiable:
Use Abs as a target function in ComplexExpand:
Solve an equation involving Abs:
Prove an inequality containing Abs:
Definite integration:
Integrate along a line in the complex plane, symbolically and numerically:
Interpret as the indefinite integral for real arguments:
Integral transforms:
Convert into Piecewise:
Denest:
ComplexPlot3D plots the magnitude of a function as height and colors using the phase:
Possible Issues (3)
Abs is a function of a complex variable and is therefore not differentiable:
As a complex function, it is not possible to write Abs[z] without involving Conjugate[z]:
In particular, the limit that defines the derivative is direction dependent and therefore does not exist:
Adding assumptions that the argument is real makes Abs differentiable:
Alternatively, use RealAbs, which assumes its argument is real:
Abs can stay unevaluated for some complicated numeric arguments:

No series can be formed from Abs for complex arguments:
For real arguments, a series can be found:
Text
Wolfram Research (1988), Abs, Wolfram Language function, https://reference.wolfram.com/language/ref/Abs.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "Abs." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Abs.html.
APA
Wolfram Language. (1988). Abs. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Abs.html