TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the magnitude or size of a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
is 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 Order or ORDER or Orders may refer to: * Orderliness, a desire for organization * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements hav ...
(or ranking)—of the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of objects to which it belongs. In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, magnitude can be defined as quantity or distance.

# History

The Greeks distinguished between several types of magnitude, including: *Positive
fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...
s *
Line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

s (ordered by
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

) * Plane figures (ordered by
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

) *
Solids Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) re ...
(ordered by
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

) *
Angles The Angles ( ang, Ængle, ; la, Angli; german: Angeln) were one of the main Germanic peoples The Germanic peoples were a historical group of people living in Central Europe and Scandinavia. Since the 19th century, they have traditional ...

(ordered by angular magnitude) They proved that the first two could not be the same, or even
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

systems of magnitude. They did not consider negative magnitudes to be meaningful, and ''magnitude'' is still primarily used in contexts in which
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

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

# Numbers

The magnitude of any
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

$x$ is usually called its ''
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

'' or ''modulus'', denoted by $, x,$.

## Real numbers

The absolute value of a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
''r'' is defined by: :$\left, r \ = r, \text r \text 0$ :$\left, r \ = -r, \text r < 0 .$ Absolute value may also be thought of as the number's
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

from zero on the real
number line In elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) include ...

. For example, the absolute value of both 70 and −70 is 70.

## Complex numbers

A
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

''z'' may be viewed as the position of a point ''P'' in a 2-dimensional space, called the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. 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 is similar to that for the
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
of a vector in a 2-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
: :$\left, z \ = \sqrt$ where the real numbers ''a'' and ''b'' are the
real part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
and the
imaginary part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
of ''z'', respectively. For instance, the modulus of is $\sqrt = 5$. Alternatively, the magnitude of a complex number ''z'' may be defined as the square root of the product of itself and its
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, $\bar$, where for any complex number $z = a + bi$, its complex conjugate is $\bar = a -bi$. :$\left, z \ = \sqrt = \sqrt = \sqrt = \sqrt$ (where $i^2 = -1$).

# Vector spaces

## Euclidean vector space

A
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
represents the position of a point ''P'' in a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. 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 coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...

s of ''P''): ''x'' = 'x''1, ''x''2, ..., ''x''''n'' Its magnitude or length, denoted by $\, x\,$, is most commonly defined as its
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
(or Euclidean length): :$\, \mathbf\, = \sqrt.$ For instance, in a 3-dimensional space, the magnitude of , 4, 12is 13 because $\sqrt = \sqrt = 13.$ This is equivalent to the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of the vector with itself: :$\, \mathbf\, = \sqrt.$ The Euclidean norm of a vector is just a special case of
Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector ''x'': #$\left \, \mathbf \right \, ,$ #$\left , \mathbf \right , .$ A disadvantage of the second notation is that it can also be used to denote the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of scalars and the
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
does not possess a magnitude. A
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
endowed with a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, such as the Euclidean space, is called a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. 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 spaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, the magnitude of a vector is the value of the
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for that vector.

# Logarithmic magnitudes

When comparing magnitudes, a
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

ic scale is often used. Examples include the
loudness In acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is ...
of a
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the b ...

(measured in
decibels The decibel (symbol: dB) is a relative unit of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...

), the
brightness Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target. It is not necessarily proportional to lumina ...

of a
star A star is an astronomical object consisting of a luminous spheroid of plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark ...

, and the
Richter scale The Richter scale – also called the Richter magnitude scale or Richter's magnitude scale – is a measure of the strength of earthquakes, developed by Charles Francis Richter and presented in his landmark 1935 paper, where he called it the "m ...
of earthquake intensity. Logarithmic magnitudes can be negative, and cannot be added or subtracted meaningfully (since the relationship is non-linear).

# 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.