This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Abs

 Abs[z]gives the absolute value of the real or complex number z.
• 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.
Real numbers:
Complex numbers:
Real numbers:
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Complex numbers:
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 Scope   (3)
Abs works with symbolic representations of numbers:
Express absolute values of algebraic numbers as explicit algebraic numbers:
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:
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:
Denest:
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:
Form nested functions involving Abs:
Plot Abs at Gaussian integers:
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