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Sqrt

Status: Stable

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

Description

Sqrt[z]
    represents the principal square root of z.
Sqrt is Listable. Sqrt[z] is canonicalised to Power[z, 1/2]; perfect
integer / rational squares reduce to exact form, negative real inputs
yield I * Sqrt[-x], and numeric inputs (Real / MPFR / Complex) are
evaluated directly. Branch cut along the negative real axis.

Examples

No verified examples yet for this function.

Implementation notes

builtin_sqrt is a thin wrapper: it rewrites Sqrt[x] to Power[x, Rational[1, 2]] (via make_rational(1, 2)) and returns that, letting the full Power machinery handle all simplification (exact perfect squares, Sqrt[8] -> 2 Sqrt[2] radical extraction, numeric/MPFR evaluation, infinity algebra). Sqrt carries LISTABLE | NUMERICFUNCTION | PROTECTED.

Attributes: Listable, NumericFunction, Protected.

Implementation status

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

References

  • Geddes, Czapor & Labahn, "Algorithms for Computer Algebra" (1992), on square-free factorization and radical simplification.
  • Knuth, "The Art of Computer Programming, Vol. 2: Seminumerical Algorithms", on integer square roots.
  • Source: src/power.c
  • Specification: docs/spec/builtins/arithmetic.md

Notes & additional examples

Worked examples

In[1]:= Sqrt[50]
Out[1]= 5 Sqrt[2]

In[1]:= Sqrt[-9]
Out[1]= 3*I

Like terms combine and products of surds simplify back to integers:

In[1]:= Sqrt[2] + Sqrt[8]
Out[1]= 3 Sqrt[2]

In[2]:= Sqrt[12]*Sqrt[27]
Out[2]= 18

Nested radicals denest under FullSimplify:

In[1]:= Sqrt[3 + 2 Sqrt[2]] // FullSimplify
Out[1]= 1 + Sqrt[2]

Principal value of a complex root, both symbolically and to 40 digits:

In[1]:= Sqrt[I]
Out[1]= (1 + I)/Sqrt[2]

In[2]:= N[Sqrt[I], 40]
Out[2]= 0.70710678118654752440084436210484903928487 + 0.70710678118654752440084436210484903928487*I

The Puiseux series of Sqrt[1 + x] is delivered exactly:

In[1]:= Series[Sqrt[1 + x], {x, 0, 5}]
Out[1]= 1 + 1/2 x - 1/8 x^2 + 1/16 x^3 - 5/128 x^4 + 7/256 x^5 + O[x]^6

Notes

Sqrt[n] is Power[n, 1/2], so it inherits the perfect-square extraction logic: the largest square factor is pulled out of the radical, reducing Sqrt[50] to 5 Sqrt[2] and Sqrt[18] to 3 Sqrt[2]. Perfect squares collapse fully, and a perfect-square rational like 1/4 yields the exact rational 1/2. Negative arguments produce a pure imaginary result, with Sqrt[-9] giving 3*I. The surd is kept in symbolic form when no square factor remains, e.g. Sqrt[2]. Denesting of compound radicals like Sqrt[3 + 2 Sqrt[2]] is not automatic but is recovered by FullSimplify. The branch cut runs along the negative real axis, so N[Sqrt[I], 40] returns the upper-half-plane principal value.