Real numbers

The absolute value of a real number r is defined by:^[4]

${\displaystyle \left|r\right|=r,{\text{ if }}r{\text{ ≥ }}0}$

${\displaystyle \left|r\right|=-r,{\text{ if }}r<0.}$

Absolute value may also be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 70 and −70 is 70.

Complex numbers

A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:^[5]

${\displaystyle \left|z\right|={\sqrt {a^{2}+b^{2}}}}$

where the real numbers a and b are the real part and the imaginary part of z, respectively. For instance, the modulus of −3 + 4i is ${\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5}$. Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, ${\displaystyle {\bar {z}}}$, where for any complex number ${\displaystyle z=a+bi}$, its complex conjugate is ${\displaystyle {\bar {z}}=a-bi}$.

${\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}-abi+abi-b^{2}i^{2}}}={\sqrt {a^{2}+b^{2}}}}$

(where ${\displaystyle i^{2}=-1}$).

Euclidean vector space

A Euclidean vector represents the position of a point P in a Euclidean space. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P): x = [x₁, x₂, ..., x_n]. Its magnitude or length, denoted by ${\displaystyle \|x\|}$,^[6] is most commonly defined as its Euclidean norm (or Euclidean length):^[7]

${\displaystyle \|\mathbf {x} \|={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}$

For instance, in a 3-dimensional space, the magnitude of [3, 4, 12] is 13 because ${\displaystyle {\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.}$ This is equivalent to the square root of the dot product of the vector with itself:

${\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}.}$

The Euclidean norm of a vector is just a special case of Euclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:

1. ${\displaystyle \left\|\mathbf {x} \right\|,}$
2. ${\displaystyle \left|\mathbf {x} \right|.}$

A disadvantage of the second notation is that it can also be used to denote the absolute value of scalars and the determinants of matrices, which introduces an element of ambiguity.

Normed vector spaces

By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude.

A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.^[8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.

Pseudo-Euclidean space

In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.