Built-in Mathematica Symbol

Abs

Abs[z]
gives the absolute value of the real or complex number z.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

For complex numbers z, Abs[z] gives the modulus .

Abs[z] is left unevaluated if z is not a numeric quantity.

Abs automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Real numbers:

Complex numbers:

Abs threads element-wise over lists:

Real numbers:

Complex numbers:

Abs threads element-wise over lists:

Scope (3)

Abs works with symbolic representations of numbers:

Express absolute values of algebraic numbers as explicit algebraic numbers:

TraditionalForm formatting:

Generalizations & Extensions (3)

Abs can deal with real-valued intervals:

Infinite arguments give symbolic results:

Abs threads element-wise over sparse arrays:

Applications (2)

Plot Abs over the complex plane:

Color plots according to Abs:

Properties & Relations (12)

Simplify expressions containing Abs:

Abs is idempotent:

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:

Obtain Abs from Limit:

Convert into Piecewise:

De-nest:

Possible Issues (3)

Norms of general vectors contain Abs:

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:

Neat Examples (2)

Form nested functions involving Abs:

Plot Abs at Gaussian integers: