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.

Notes

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 (9)¶

Every input below was run against the current Mathilda build and its output recorded.

Worked examples (1)¶

In[1]:= Sqrt[72]
Out[1]= 6 Sqrt[2]

Applications (8)¶

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

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

In[4]:= Sqrt[2] + Sqrt[8]
Out[4]= 3 Sqrt[2]

In[5]:= Sqrt[12]*Sqrt[27]
Out[5]= 18

In[6]:= Sqrt[3 + 2 Sqrt[2]] // FullSimplify
Out[6]= 1 + Sqrt[2]

In[7]:= Sqrt[I]
Out[7]= (1 + I)/Sqrt[2]

In[8]:= N[Sqrt[I], 40]
Out[8]= 0.70710678118654752440084436210484903928487 + 0.70710678118654752440084436210484903928487*I

In[9]:= Series[Sqrt[1 + x], {x, 0, 5}]
Out[9]= 1 + 1/2 x - 1/8 x^2 + 1/16 x^3 - 5/128 x^4 + 7/256 x^5 + O[x]^6

Performance¶

Against other systems, from the benchmark suite (same input, results cross-checked for agreement):

case	Mathilda	Wolfram	Python
NI 50-digit Gaussian	6.87 s	2.11 s	1.1 s
NI 2-D ridge	2.54 s	43.5 s	0.685 s
NI oscillatory k=40	0.515 s	4.05 s	0.002 s
NI oscillatory k=200	0.477 s	21.5 s	0.002 s
NI oscillatory k=1000	0.389 s	140 s	0.002 s
NI oscillatory k=1001 nonzero	0.382 s	1.03 s	0.626 s

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.

References¶

See also: FactorInteger

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
Tests: tests/test_airyai.c
Tests: tests/test_airybi.c
Tests: tests/test_algebraicnumber.c
Tests: tests/test_algebraicnumberdenominator.c

Notes & additional examples¶

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.