Magnitude (mathematics)

In

of a number is commonly applied as the measure of units between a number and zero.

In vector spaces, the

is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.

History

including:

• Positive fractions
• Line segments (ordered by length)
• Plane figures (ordered by area
  )
• Solids (ordered by volume)
• Angles (ordered by angular magnitude)

They proved that the first two could not be the same, or even

is either the smallest size or less than all possible sizes.

Numbers

The magnitude of any number ${\displaystyle x}$ is usually called its absolute value or modulus, denoted by ${\displaystyle |x|}$.^[3]

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

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

where the real numbers a and b are 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 ${\displaystyle i^{2}=-1}$).

Vector spaces

Euclidean vector space

A

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.

Logarithmic magnitudes

When comparing magnitudes, a

.

Order of magnitude

Orders of magnitude denote differences in numeric quantities, usually measurements, by a factor of 10—that is, a difference of one digit in the location of the decimal point.

Other mathematical measures

See also

• Number sense
• Vector notation
• Set size

References