Sign

The sign of x: -1, 0, or 1 for negative, zero, or positive x.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSign function↗MathWorldSign↗WikidataQ236813↗DLMFfront/introduction.common.t2.r18↗FungrimSign✓↗Fungrim entryFindStatSt000037via Permutation✓↗Wolfram LanguageSign!↗

Sign(x)-1, 0, or 1 for negative, zero, or positive real x; $z/|z|$ for complex z.

SageMathenumeratio_sign(x)SymPysign(x)↗mpmathsign(x)↗Oscarsign(x)Juliaenumeratio_sign(x)Rustsignum(x)

Sign(p)A permutation's sign: 1 when it has an even number of inversions, -1 when odd (FindStat St000037).

SageMathenumeratio_sign(p)SymPysign(p)↗mpmathsign(p)↗Oscarsign(p)Juliaenumeratio_sign(p)Rustsignum(p)

Typed signatures

Type	Package	Applies
(complex | permutation | signed_infinity | ~oo) -> complex | number	combinatoricsreplaces analytic	always
(complex | signed_infinity | ~oo) -> complex	analyticreplaces compute-engine	on Around, CenteredInterval, Interval
(complex | signed_infinity | ~oo) -> complex	compute-engine	always

Domain: Arithmetic

Details

• $\operatorname{sign}(x) = -1, 0, 1$ for $x < 0$, $x = 0$, $x > 0$ respectively.
• For a complex number, $\operatorname{sign}(z) = z/|z|$, the unit complex number pointing toward z. See Abs.
• Recovers the original magnitude: $x = |x|\,\operatorname{sign}(x)$.
• $\operatorname{sign}(\pm\infty) = \pm 1$, but $\operatorname{sign}(\mathrm{NaN})$ propagates as NaN rather than 0.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

1 if Length(_x) < 2 else (-1) ^ Sum(Map(
                                         Function(Count(Filter(
                                                                Range(
                                                                       i + 1,
                                                                       Length(_x),
                                                                     ),
                                                                Function(_x[i] >
                                                                         _x[j], j),
                                                              )), i),
                                         Range(1, Length(_x)),
                                       ))

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\mathrm{sgn}(-3)$Out$-1$

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In$\mathrm{sgn}(3)$Out$1$

Zero is its own sign

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In$\mathrm{sgn}(0)$Out$0$

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In$\mathrm{sgn}(-\frac{1}{2})$Out$-1$

The unit complex number pointing the same direction as $3+4i$

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In$\mathrm{sgn}(3+4i)$Out$\frac{3}{5}+\frac{4}{5}i$

Threads element-wise over a list

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In$\mathrm{sgn}(\bigl\lbrack-3, 0, 7\bigr\rbrack)$Out$\bigl\lbrack-1, 0, 1\bigr\rbrack$

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In$\mathrm{sgn}(-2.5)$Out$-1$

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In$\mathrm{sgn}(3.14)$Out$1$

A complex number with approximate parts

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In$\mathrm{sgn}(1.4+2.3i)$Out$0.519\,946\,946\,895\,745\,216\,810\,2+0.854\,198\,555\,614\,438\,5i$

Scope #

An exact complex argument comes back exact: $\mathrm{sign}(1+i) = (1+i)/\sqrt{2}$

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In$\mathrm{sgn}(1+i)$Out$\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$

Threads over a matrix, entry by entry

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In$\mathrm{sgn}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack-\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack, \bigl\lbrack-1, 1\bigr\rbrack\bigr\rbrack$

Exact numeric constants

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In$\mathrm{sgn}(π)$Out$1$

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In$\mathrm{sgn}(-e)$Out$-1$

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In$\mathrm{sgn}(i)$Out$i$

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In$\mathrm{sgn}(∞)$Out$1$

An Interval entirely above 0 has sign 1

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In$\mathrm{sgn}(\mathrm{Interval}(1, 3))$Out$1$

Decides the sign of the exact number $\sqrt{2}-2$ by certified bignum evaluation at increasing precision

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

$2-\sqrt{2}$ is the same certified decision, the other way

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

A positive number has sign 1

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In$\mathrm{sgn}(123)$Out$1$

For a complex number it is $z/|z|$, here with a radical

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In$\mathrm{sgn}(123-456i)$Out$\frac{41\sqrt{24\,785}}{24\,785}-\frac{152\sqrt{24\,785}}{24\,785}i$

A matrix function with exact eigenvalues, giving a zero diagonal

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{Sign}, \bigl\lbrack\bigl\lbrack1/2, -1\bigr\rbrack, \bigl\lbrack-5/3, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack0, -\frac{15^{1/2}}{5}\bigr\rbrack, \bigl\lbrack-\frac{15^{1/2}}{3}, 0\bigr\rbrack\bigr\rbrack$

On $(-2,2)$ it is a piecewise function with 0 at the origin

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In$\mathrm{PiecewiseExpand}(\mathrm{sgn}(x), -2\lt x\lt2)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack-1, x\lt0\bigr\rbrack, \bigl\lbrack1, 0\lt x\bigr\rbrack\bigr\rbrack, 0)$

Sign commutes with conjugation

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In$\mathrm{sgn}(\overline{1+2i})=\overline{\mathrm{sgn}(1+2i)}$Out$\mathrm{True}$

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In$\frac{\mathrm{d}\mathrm{sgn}(x)}{\mathrm{d}x}$Out$0$

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In$\mathrm{sgn}(∞i)$Out$\operatorname{Indeterminate}$

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In$N(\mathrm{sgn}(e+i π), 20)$Out$0.654\,321\,120\,736\,689\,028\,19+0.756\,216\,814\,781\,239\,322\,47i$

Applications #

Tells you the temperature dropped, but not by how much -- for that, see Abs

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In$\mathrm{sgn}(70-95)$Out$-1$

Properties #

$\operatorname{sign}(z) = z/|z|$. See Abs

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In$\mathrm{sgn}(3+4i)=\frac{3+4i}{\vert3+4i\vert}$Out$\mathrm{True}$

$x \cdot \operatorname{sign}(x) = |x|$, here at $x = -5$

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In$-5\mathrm{sgn}(-5)=\vert-5\vert$Out$\mathrm{True}$

The sign of a nonzero complex number lies on the unit circle

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In$\vert\mathrm{sgn}(3+4i)\vert=1$Out$\mathrm{True}$

Positive 1 has sign 1

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In$\mathrm{sgn}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, Sign

Negative 1 gives $-1$

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In$\mathrm{sgn}(-1)$Out$-1$

Known value · Wolfram MathematicalFunctionData, Sign

A unit imaginary stays $-i$

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In$\mathrm{sgn}(-i)$Out$-i$

Known value · Wolfram MathematicalFunctionData, Sign

The second quadrant gives $(-1+i)/\sqrt2$

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In$\mathrm{sgn}(-1+i)$Out$-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$

Known value · Wolfram MathematicalFunctionData, Sign

The third quadrant gives the unit vector at $225^\circ$

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In$\mathrm{sgn}(-1-i)$Out$-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$

Known value · Wolfram MathematicalFunctionData, Sign

The fourth quadrant, $(1-i)/\sqrt2$

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In$\mathrm{sgn}(1-i)$Out$\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$

Known value · Wolfram MathematicalFunctionData, Sign

Positive integers always map to 1

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In$\mathrm{sgn}(2)$Out$1$

Known value · Wolfram MathematicalFunctionData, Sign

FindStat's value for [1]

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In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{Sign})$Out$1$

Known value · FindStat St000037

FindStat's value for [1,2]

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In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{Sign})$Out$1$

Known value · FindStat St000037

FindStat's value for [2,1]

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In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{Sign})$Out$-1$

Known value · FindStat St000037

Possible issues #

NaN propagates rather than being treated as 0

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In$\mathrm{sgn}(\operatorname{NaN})$Out$\operatorname{NaN}$

Sign(ComplexInfinity) is Indeterminate: z/|z| needs a direction, and the undirected infinity has none

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In$\mathrm{sgn}(\tilde∞)$Out$\operatorname{Indeterminate}$

Known value · Wolfram Sign[ComplexInfinity]

About

Examples	43
Checked against	Wolfram Language 38 agree2 differSageMath 36 agree2 differ2 unevaluatedSymPy 35 agree2 differ2 unevaluatedmpmath 35 agreeOscar 22 agreeJulia 33 agree2 errorRust 18 agree
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×23FindStatWolfram LanguageSageMathSymPympmathOscarJuliaRust

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Abs, Negate