Sqrt

The principal square root of x: $\sqrt{x}$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSquare root↗MathWorldSquareRoot↗WikidataQ134237↗Britannicatopic/square-root↗FungrimSqrt↗Wolfram LanguageSqrt!↗

Sqrt(x)the principal square root $\sqrt{x} = x^{1/2}$.

SageMathsqrt(x)✓↗SymPysqrt(x)↗mpmathsqrt(x)↗Rustsqrt(x)

Typed signatures

Type	Package	Applies
(complex | infinity) -> complex | infinity	analyticreplaces compute-engine	on Around, CenteredInterval, Interval
(complex | infinity) -> complex | infinity	compute-engine	always

Domain: Arithmetic

Details

• The principal square root: $\sqrt{x} = x^{1/2}$. See Root for other roots and Square for the inverse operation.
• Evaluates exactly only when the radicand is a perfect square (or a ratio of perfect squares); otherwise it stays in symbolic surd form, like $\sqrt{2}$.
• A negative radicand gives an imaginary result, $\sqrt{-a} = i\sqrt{a}$ for $a>0$ -- but only reduces to an exact Complex number when $a$ itself is a perfect square.
• $\sqrt{0} = 0$.
• compute-engine's Sqrt is a special case of the more general Root: $\mathrm{Root}(x, 2)$ canonicalizes to $\mathrm{Sqrt}(x)$.

Examples

8 more kept as data, checked against the oracles

Basic #

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In$\sqrt{16}$Out$4$

Stays symbolic when the radicand isn't a perfect square

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In$\sqrt{2}$Out$\sqrt{2}$

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In$\sqrt{\frac{4}{9}}$Out$\frac{2}{3}$

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In$\sqrt{0}$Out$0$

A perfect square radicand evaluates exactly, even when negative

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In$\sqrt{-4}$Out$2i$

An approximate number gives an approximate root

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In$\sqrt{3.5}$Out$1.870\,828\,693\,386\,970\,692\,79$

Pulls perfect-square factors out of the radical

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In$\sqrt{200}$Out$10\sqrt{2}$

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In$\sqrt{-25}$Out$5i$

Scope #

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In$\sqrt{8}$Out$2\sqrt{2}$

Rationalizes the denominator

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In$\sqrt{\frac{1}{2}}$Out$\frac{\sqrt{2}}{2}$

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In$\sqrt{\frac{12}{5}}$Out$\frac{2\sqrt{15}}{5}$

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In$\sqrt{-2.5}$Out$1.581\,138\,830\,084\,189\,8i$

An exact complex square: $(2+i)^2 = 3+4i$

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In$\sqrt{3+4i}$Out$2+i$

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In$\sqrt{∞}$Out$∞$

Sqrt(-Infinity) = i Infinity, the principal branch's direction as the radicand goes to -Infinity (DLMF 4.2.2)

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In$\sqrt{-∞}$Out$\tilde∞$

Known value · Wolfram MathematicalFunctionData, Sqrt

Threads element-wise over a list

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In$\sqrt{\bigl\lbrack1, 4, 9\bigr\rbrack}$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Threads over a matrix entry by entry -- this is not the matrix square root

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In$\sqrt{\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack}$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{2}}{2}, i\bigr\rbrack, \bigl\lbrack0, \frac{\sqrt{2}}{2}\bigr\rbrack\bigr\rbrack$

Interval arithmetic: $\sqrt{}$ is increasing, so $\sqrt{[1,8]} = [1, 2\sqrt2]$

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In$\sqrt{\mathrm{Interval}(1, 8)}$Out$\mathrm{Interval}(1, 2\sqrt{2})$

To 45 significant digits, every one correctly rounded

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In$N(\sqrt{2}, 45)$Out$1.414\,213\,562\,373\,095\,048\,801\,688\,724\,209\,698\,078\,569\,671\,88$

Applications #

The Pythagorean hypotenuse of a 3-4-5 triangle

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In$\sqrt{3^2+4^2}$Out$5$

Properties #

$(\sqrt{x})^2 = x$ for a perfect square. See Square

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In$\sqrt{16}^2=16$Out$\mathrm{True}$

$\sqrt{x} = x^{1/2}$

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In$\sqrt{16}=\sqrt{16}$Out$\mathrm{True}$

$(\sqrt{x})^2 = x$ for every x

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In$\sqrt{x}^2$Out$x$

Possible issues #

Negative but not a perfect square: stays symbolic rather than reducing to an exact complex value

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In$\sqrt{-2}$Out$\sqrt{-2}$

A negative rational still evaluates exactly when both numerator and denominator are perfect squares

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In$\sqrt{-\frac{4}{9}}$Out$\frac{2}{3}i$

$\sqrt{x^2}$ is not simplified to x, which would be wrong for negative x

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In$\sqrt{x^2}$Out$\sqrt{x^2}$

...at $x = -1$, $\sqrt{x^2} = 1$, not $-1$

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In$\sqrt{(-1)^2}$Out$1$

$\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails for negative a and b: $\sqrt{-1}\sqrt{-1} = -1$...

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In$\sqrt{-1}\sqrt{-1}$Out$-1$

...while $\sqrt{(-1)(-1)} = 1$

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In$\sqrt{1}$Out$1$

About

Examples	37
Checked against	Wolfram Language 36 agree1 differSageMath 33 agreeSymPy 35 agree1 differmpmath 31 agree1 differRust 13 agree12 differ1 unevaluated2 error
Crosswalk	WikipediaMathWorldWikidataBritannicaFungrimWolfram LanguageSageMathSymPympmathRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Root, Square