In

2, ..., ''x''''n''">'x''_{1}, ''x''_{2}, ..., ''x''_{''n''} Its magnitude or length, denoted by $\backslash ,\; x\backslash ,$, is most commonly defined as its

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: *Positivefraction
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 anynumber
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 areal 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:
:$\backslash left,\; r\; \backslash \; =\; r,\; \backslash text\; r\; \backslash text\; 0$
:$\backslash left,\; r\; \backslash \; =\; -r,\; \backslash text\; r\; <\; 0\; .$
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

Acomplex 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 ...

:
:$\backslash left,\; z\; \backslash \; =\; \backslash sqrt$
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 $\backslash sqrt\; =\; 5$. 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 ...

, $\backslash bar$, where for any complex number $z\; =\; a\; +\; bi$, its complex conjugate is $\backslash bar\; =\; a\; -bi$.
:$\backslash left,\; z\; \backslash \; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt$
(where $i^2\; =\; -1$).
Vector spaces

Euclidean vector space

AEuclidean 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'' = 1, ''x''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):
:$\backslash ,\; \backslash mathbf\backslash ,\; =\; \backslash sqrt.$
For instance, in a 3-dimensional space, the magnitude of , 4, 12is 13 because $\backslash sqrt\; =\; \backslash sqrt\; =\; 13.$
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:
:$\backslash ,\; \backslash mathbf\backslash ,\; =\; \backslash sqrt.$
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'':
#$\backslash left\; \backslash ,\; \backslash mathbf\; \backslash right\; \backslash ,\; ,$
#$\backslash left\; ,\; \backslash mathbf\; \backslash right\; ,\; .$
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 abstractvector 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 apseudo-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, alogarithm
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.See also

*Number senseIn psychology
Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scope ...

*Vector notation
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 ...

* Set size
References

{{reflist Elementary mathematics Unary operations