RealSign

The sign of a real number x: -1, 0 or 1, as a function defined only on the reals.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageRealSign!↗

RealSign(x)-1, 0 or 1, according to the sign of the real number x.

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Elementary functions

Details

• Restricted to the reals, the sign is always decidable, so RealSign always resolves to an exact -1, 0 or 1 -- unlike native Sign, which stays symbolic at an exact expression it cannot immediately classify (e.g. $\mathrm{Sign}(\sqrt2 - 2)$ stays unevaluated under plain evaluation).
• A concretely complex argument is outside RealSign's domain and is left unevaluated.

Plot

Examples

Basic #

live ↗

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

live ↗

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

live ↗

In$\mathrm{RealSign}(2.5)$Out$1$

A positive number has sign 1

live ↗

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

Scope #

Threads element-wise over a list

live ↗

In$\mathrm{RealSign}(\bigl\lbrack-2, 0, 5\bigr\rbrack)$Out$\bigl\lbrack-1, 0, 1\bigr\rbrack$

Decides the sign of an exact numeric expression

live ↗

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

Entrywise sign of a matrix

live ↗

In$\mathrm{RealSign}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, -\frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

As a matrix function it acts on eigenvalues, which changes the off-diagonal to $-2$

live ↗

In$\mathrm{MatrixFunction}(\mathrm{RealSign}, \bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, -\frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, -2\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

The symbol stays, as the sign depends on $x$

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{RealSign}(x))$Out$\mathrm{RealSign}(x)$

Properties #

Real entries evaluate; the complex entry is left, since RealSign is real-only

live ↗

In$\mathrm{RealSign}(\bigl\lbrack3, -5, 2+5i\bigr\rbrack)$Out$\bigl\lbrack1, -1, \mathrm{RealSign}(2+5i)\bigr\rbrack$

About

Examples	10
Checked against	Wolfram Language 9 agree1 differ
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