Inmathematics, themagnitude orsize of amathematical object 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 anordering (or ranking) of theclass of objects to which it belongs. Magnitude as a concept dates toAncient Greece and has been applied as ameasure of distance from one object to another. For numbers, theabsolute value of a number is commonly applied as the measure of units between a number and zero.
In vector spaces, theEuclidean norm is a measure of magnitude used to define a distance between two points in space. Inphysics, magnitude can be defined as quantity or distance. Anorder of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.
Ancient Greeks distinguished between several types of magnitude,[1] including:
They proved that the first two could not be the same, or evenisomorphic systems of magnitude.[2] They did not considernegative magnitudes to be meaningful, andmagnitude is still primarily used in contexts in whichzero is either the smallest size or less than all possible sizes.
The magnitude of anynumber is usually called itsabsolute value ormodulus, denoted by.[3]
The absolute value of areal numberr is defined by:[4]
Absolute value may also be thought of as the number'sdistance from zero on the realnumber line. For example, the absolute value of both 70 and −70 is 70.
Acomplex numberz may be viewed as the position of a pointP in a2-dimensional space, called thecomplex plane. The absolute value (ormodulus) ofz may be thought of as the distance ofP from the origin of that space. The formula for the absolute value ofz =a +bi is similar to that for theEuclidean norm of a vector in a 2-dimensionalEuclidean space:[5]
where the real numbersa andb are thereal part and theimaginary part ofz, respectively. For instance, the modulus of−3 + 4i is. Alternatively, the magnitude of a complex numberz may be defined as the square root of the product of itself and itscomplex conjugate,, where for any complex number, its complex conjugate is.
(where).
AEuclidean vector represents the position of a pointP in aEuclidean space. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vectorx in ann-dimensional Euclidean space can be defined as an ordered list ofn real numbers (theCartesian coordinates ofP):x = [x1,x2, ...,xn]. Itsmagnitude orlength, denoted by,[6] is most commonly defined as itsEuclidean norm (or Euclidean length):[7]
For instance, in a 3-dimensional space, the magnitude of [3, 4, 12] is 13 becauseThis is equivalent to thesquare root of thedot product of the vector with itself:
The Euclidean norm of a vector is just a special case ofEuclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vectorx:
A disadvantage of the second notation is that it can also be used to denote theabsolute value ofscalars and thedeterminants ofmatrices, which introduces an element of ambiguity.
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstractvector space does not possess a magnitude.
Avector space endowed with anorm, such as the Euclidean space, is called anormed vector space.[8] The norm of a vectorv in a normed vector space can be considered to be the magnitude ofv.
In apseudo-Euclidean space, the magnitude of a vector is the value of thequadratic form for that vector.
When comparing magnitudes, alogarithmic scale is often used. Examples include theloudness of asound (measured indecibels), thebrightness of astar, and theRichter scale of earthquake intensity. Logarithmic magnitudes can be negative. In thenatural sciences, a logarithmic magnitude is typically referred to as alevel.
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.
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Inmathematics, the concept of ameasure is a generalization and formalization ofgeometrical measures (length,area,volume) and other common notions, such as magnitude,mass, andprobability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational inprobability theory,integration theory, and can be generalized to assumenegative values, as withelectrical charge. Far-reaching generalizations (such asspectral measures andprojection-valued measures) of measure are widely used inquantum physics and physics in general.
The intuition behind this concept dates back toAncient Greece, whenArchimedes tried to calculate thearea of a circle.[9][10] But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works ofÉmile Borel,Henri Lebesgue,Nikolai Luzin,Johann Radon,Constantin Carathéodory, andMaurice Fréchet, among others. According toThomas W. Hawkins Jr., "It was primarily through the theory ofmultiple integrals and, in particular the work ofCamille Jordan that the importance of the notion of measurability was first recognized."[11]The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece.