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Magnitude (mathematics)

In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs.

History

The Greeks distinguished between several types of magnitude,[1] including:

They proved that the first two could not be the same, or even isomorphic systems of magnitude.[2] They did not consider negative magnitudes to be meaningful, and magnitude is still chiefly used in contexts in which zero is either the smallest size or less than all possible sizes.

Numbers

The magnitude of any number is usually called its "absolute value" or "modulus", denoted by |x|.

Real numbers

The absolute value of a real number r is defined by:[3]

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

Absolute value may 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:[4]

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

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, z, where for any complex number z = a + bi, its complex conjugate is z = abi.

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

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

Vector spaces

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, ..., xn]. Its magnitude or length is most commonly defined as its Euclidean norm (or Euclidean length):[5]

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

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 by 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 is also used to denote the absolute value of scalars and the determinants of matrices and therefore can be ambiguous.

Normed vector spaces

By definition, all Euclidean vectors have a magnitude (see above). However, the notion of magnitude cannot be applied to all kinds of vectors.

A function that maps objects to their magnitudes is called a norm. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.[6] Not all vector spaces are normed.

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 logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. It is not meaningful to simply add or subtract them.

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.

References

1. ^ Heath, Thomas Smd. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.
2. ^ Bloch, Ethan D. (2011), The Real Numbers and Real Analysis, Springer, p. 52, ISBN 9780387721774, The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece.
3. ^ Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2.
4. ^ Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha.
5. ^ Howard Anton; Chris Rorres (12 April 2010). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-0-470-43205-1.
6. ^ Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 978-1-4020-5494-5