Abs

The absolute value or magnitude of x: $|x|$, its distance from 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaAbsolute value↗Absolute value of a complex number↗MathWorldAbsoluteValue↗ComplexModulus↗WikidataQ3317982↗Britannicatopic/absolute-value↗DLMF1.9.E7↗FungrimAbs✓↗Fungrim entryWolfram LanguageAbs!↗

Abs(x)the absolute value $|x|$ of a real x, or the modulus of a complex number.

SageMathenumeratio_abs(x)SymPyAbs(x)↗mpmathfabs(x)↗Oscarabs(x)Juliaenumeratio_ball(abs, x)Rustabs(x)

Typed signatures

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

Domain: Arithmetic

Details

• For a real number, $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.
• For a complex number $a+bi$, $|a+bi| = \sqrt{a^2+b^2}$, its distance from the origin.
• Always non-negative, with $|x| = 0$ exactly at $x = 0$.
• Satisfies the triangle inequality $|a+b| \le |a| + |b|$.
• compute-engine doesn't factor constants out of a symbolic argument -- $|-3x|$ stays as written rather than simplifying to $3|x|$.

Examples

2 more kept as data, checked against the oracles

Basic #

live ↗

In$\vert-5\vert$Out$5$

live ↗

In$\vert5\vert$Out$5$

0 is the only value with $|x| = 0$

live ↗

In$\vert0\vert$Out$0$

live ↗

In$\vert-\frac{7}{2}\vert$Out$\frac{7}{2}$

The modulus of a complex number: $|3+4i| = \sqrt{3^2+4^2}$

live ↗

In$\vert3+4i\vert$Out$5$

Threads element-wise over a list

live ↗

In$\vert\bigl\lbrack-1, 2, -3\bigr\rbrack\vert$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

live ↗

In$\vert-2.5\vert$Out$2.5$

The modulus of a complex number with approximate parts

live ↗

In$\vert1.4+2.3i\vert$Out$2.692\,582\,403\,567\,252\,015\,63$

Scope #

Threads over a matrix, entry by entry

live ↗

In$\vert\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack-\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack\vert$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}, 1\bigr\rbrack, \bigl\lbrack\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack$

Exact numeric constants

live ↗

In$\vert-π\vert$Out$π$

An exact complex number gives an exact radical

live ↗

In$\vert1+i\vert$Out$\sqrt{2}$

live ↗

In$\vert i\vert$Out$1$

$e^{i\pi/3}$ lies on the unit circle

live ↗

In$\vert e^{\frac{i π}{3}}\vert$Out$1$

live ↗

In$\vert-∞\vert$Out$∞$

Complex infinity has infinite magnitude in every direction

live ↗

In$\vert\tilde∞\vert$Out$∞$

An Interval maps to the interval of absolute values, $[0, 5]$

live ↗

In$\vert\mathrm{Interval}(-3, 5)\vert$Out$\mathrm{Interval}(0, 5)$

Decides the sign of an exact numeric expression and gives $2-\sqrt{2}$

live ↗

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

Twenty-five digits of the modulus of a complex constant

live ↗

In$N(\vert π+i\operatorname{G}\vert, 25)$Out$3.272\,399\,329\,361\,528\,960\,349\,014$

Applied to a matrix as a matrix function, not entry by entry

live ↗

In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{Abs}, \bigl\lbrack\bigl\lbrack1/2, -1\bigr\rbrack, \bigl\lbrack-5/3, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{15}}{3}, -\frac{15^{1/2}}{10}\bigr\rbrack, \bigl\lbrack-\frac{15^{1/2}}{6}, \frac{\sqrt{15}}{3}\bigr\rbrack\bigr\rbrack$

The modulus of an infinite real is infinite

live ↗

In$\vert∞\vert$Out$∞$

A directed infinity still has infinite modulus

live ↗

In$\vert∞i\vert$Out$∞$

With the assumption on $x$, the call still stays unexpanded

live ↗

In$\mathrm{PiecewiseExpand}(\vert i x\vert, -2\lt x\lt2)$Out$\vert i x\vert$

Treating $x$ as real, $|x| = \sqrt{x^2}$

live ↗

In$\mathrm{ComplexExpand}(\vert x\vert)$Out$\sqrt{x^2}$

Real and imaginary parts give $|x+iy| = \sqrt{x^2+y^2}$

live ↗

In$\mathrm{ComplexExpand}(\vert x+i y\vert)$Out$\sqrt{x^2+y^2}$

live ↗

In$\mathrm{Abs}^{\prime}(x)$Out$\mathrm{sgn}(x)$

Applications #

The gap between two temperatures, 70° and 95°, regardless of which is higher

live ↗

In$\vert70-95\vert$Out$25$

The distance between the points $1+i$ and $4+5i$ of the complex plane

live ↗

In$\vert(1+i)-(4+5i)\vert$Out$5$

Properties #

Even function: $|-x| = |x|$

live ↗

In$\vert-8\vert=\vert8\vert$Out$\mathrm{True}$

Triangle inequality: $|a+b| \le |a| + |b|$

live ↗

In$\vert3-8\vert≤\vert3\vert+\vert-8\vert$Out$\mathrm{True}$

$|z|^2 = z\bar{z}$

live ↗

In$\vert3+4i\vert^2=(3+4i)\overline{3+4i}$Out$\mathrm{True}$

Multiplicative: $|ab| = |a|\,|b|$

live ↗

In$\vert(1+2i)(3-i)\vert=\vert1+2i\vert\vert3-i\vert$Out$\mathrm{True}$

A positive real is its own modulus

live ↗

In$\vert1\vert$Out$1$

Known value · Wolfram MathematicalFunctionData, Abs

A negative real flips sign

live ↗

In$\vert-1\vert$Out$1$

Known value · Wolfram MathematicalFunctionData, Abs

A pure imaginary unit has modulus 1

live ↗

In$\vert-i\vert$Out$1$

Known value · Wolfram MathematicalFunctionData, Abs

The Gaussian integer $-1+i$ has modulus $\sqrt2$

live ↗

In$\vert-1+i\vert$Out$\sqrt{2}$

Known value · Wolfram MathematicalFunctionData, Abs

Same modulus in the third quadrant

live ↗

In$\vert-1-i\vert$Out$\sqrt{2}$

Known value · Wolfram MathematicalFunctionData, Abs

Same modulus in the fourth quadrant

live ↗

In$\vert1-i\vert$Out$\sqrt{2}$

Known value · Wolfram MathematicalFunctionData, Abs

The $30^\circ$ right triangle gives exact modulus 2

live ↗

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

Known value · Wolfram MathematicalFunctionData, Abs

Swapping the legs still gives 2

live ↗

In$\vert1+\sqrt{3}i\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Second quadrant, still 2

live ↗

In$\vert\sqrt{3}i-1\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Second quadrant with the legs swapped

live ↗

In$\vert i-\sqrt{3}\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Third quadrant, still 2

live ↗

In$\vert-i-\sqrt{3}\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Written with nested negations, still exactly 2

live ↗

In$\vert-\sqrt{3}i-1\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Fourth quadrant with a negated imaginary part, still 2

live ↗

In$\vert1-\sqrt{3}i\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

Fourth quadrant with the legs swapped

live ↗

In$\vert\sqrt{3}-i\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

A positive integer is its own modulus

live ↗

In$\vert2\vert$Out$2$

Known value · Wolfram MathematicalFunctionData, Abs

live ↗

In$\vert\frac{1}{2}\vert^2$Out$\frac{1}{4}$

Known value · Wolfram MathematicalFunctionData, Abs, Absolute square

live ↗

In$\vert2\vert^2$Out$4$

Known value · Wolfram MathematicalFunctionData, Abs, Absolute square

Possible issues #

Doesn't factor constants out of a symbolic argument: $|-3x|$ stays as written rather than reducing to $3|x|$

live ↗

In$\vert-3x\vert$Out$\vert-3x\vert$

About

Examples	51
Checked against	Wolfram Language 48 agree2 differ1 unevaluatedSageMath 40 agree2 differSymPy 46 agree1 differmpmath 23 agree1 differOscar 20 agreeJulia 37 agree3 errorRust 17 agree1 error
Crosswalk	Wikipedia ×2MathWorld ×2WikidataBritannicaDLMFFungrimFungrim entry ×57Wolfram LanguageSageMathSymPympmathOscarJuliaRust

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Sign, Chop