Absolute value

The graph of the absolute value function for real numbers
The absolute value of a number may be thought of as its distance from zero.
Composition of absolute value with a cubic function in different orders

3, and the absolute value of −3 is also 3.

- Absolute value

254 related topics


Complex number

[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number can be visually represented as a pair of numbers

A complex number z, as a point (black) and its position vector (blue)
Argument φ and modulus r locate a point in the complex plane.
Geometric representation of z and its conjugate in the complex plane
Addition of two complex numbers can be done geometrically by constructing a parallelogram.
The Mandelbrot set with the real and imaginary axes labeled.
Construction of a regular pentagon using straightedge and compass.
Cayley Q8 quaternion graph showing cycles of multiplication by, and

The complex numbers of absolute value one form the unit circle.

Norm (mathematics)

Function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.

Illustrations of unit circles in different norms.

The absolute value

Vertical bar

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The code point 124 (7C hexadecimal) is occupied by a broken bar in a dot matrix printer of the late 1980s, which apparently lacks a solid vertical bar. See the [[:Image:Dot printer ASCII.png|full picture]].
US International keyboard showing broken bar on the keycap, even though typing shift+that key produces the solid vertical bar.
Full character set of IBM's Code page 437 rendered in VGA, which displays the broken bar glyph for codepoint 7C, despite the 1977 revision to ASCII

absolute value: . See below about LaTeX in text mode.

Euclidean distance

Length of a line segment between the two points.

Using the Pythagorean theorem to compute two-dimensional Euclidean distance

The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates.

Magnitude (mathematics)

Property which determines whether the object is larger or smaller than other objects of the same kind.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space.

Holomorphic function

Complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space

A rectangular grid (top) and its image under a conformal map f (bottom).

Therefore, the absolute value

Matrix (mathematics)


Schematic depiction of the matrix product AB of two matrices A and B.
The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.
A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.
An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.
An undirected graph with adjacency matrix:
Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by (black).

Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.


Scalar value that is a function of the entries of a square matrix.

The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors.

Even and odd functions

Even function if n is an even integer, and it is an odd function if n is an odd integer.

is neither even nor odd.

If a function is odd, the absolute value of that function is an even function.

Absolute difference

Showing the absolute difference of real numbers x and y as the distance between them on the real line.

The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference.