Skip to content

Abs

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Abs[z] gives the absolute value (modulus) of numeric z, Sqrt[Re[z]^2 + Im[z]^2] for complex z.

Examples

No verified examples yet for this function.

Implementation notes

builtin_abs handles each numeric kind directly: mpz_abs for EXPR_BIGINT, sign flips for EXPR_INTEGER/EXPR_REAL/EXPR_MPFR (via mpfr_abs), and |n|/d for rationals. For a Complex[re, im] literal (or an expression complex_decompose splits into numeric real/imag parts) it builds the symbolic modulus Power[Plus[re^2, im^2], 1/2]; when MPFR components are present it folds directly through mpfr_hypot at the combined working precision instead, which is also numerically stable across disparate magnitudes. Symbolic arguments return NULL.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Abs[-5]
Out[1]= 5

In[2]:= Abs[3 + 4 I]
Out[2]= 5

In[3]:= Abs[-3/4]
Out[3]= 3/4

In[4]:= Abs[{-1, 2, -3}]
Out[4]= {1, 2, 3}

In[1]:= Abs[(1 + I)^10]
Out[1]= 32

In[2]:= Abs[Sqrt[2] + Sqrt[3] I]
Out[2]= Sqrt[5]

In[3]:= N[Abs[Gamma[1/3 + 2 I]], 30]
Out[3]= 0.09665959425732664141022797859867

Notes

For complex z, Abs[z] returns the modulus Sqrt[Re[z]^2 + Im[z]^2]. Abs is Listable, so it threads element-wise over lists. Exact arguments give exact results: Abs[(1 + I)^10] is 32 (since |1 + I| = Sqrt[2] and (Sqrt[2])^10 = 32), and Abs[Sqrt[2] + Sqrt[3] I] collapses to Sqrt[5]. The modulus of a complex value of a special function such as Gamma is evaluated to the requested precision under MPFR.