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In mathematics, '''magnitude''' is the size of a [[mathematical object]], a property by which the object can be compared as larger or smaller than other objects of the same kind. More formally, an object's magnitude is an [[order theory|ordering]] (or ranking) of the [[class (mathematics)|class]] of objects to which it belongs.
 
==History==
The Greeks distinguished between several types of magnitude,{{citation needed|date=January 2012}}  including:
*Positive [[fractions]]
*[[Line segment]]s (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 [[isomorphic]] systems of magnitude.{{citation needed|date=January 2012}} They did not consider negative magnitudes to be meaningful, and ''magnitude'' is still chiefly used in contexts in which zero is either the lowest size or less than all possible sizes.
 
==Numbers==
{{Main|Absolute value}}
The magnitude of any [[number]] ''x'' is usually called its "[[absolute value]]" or "modulus", denoted by |''x''|.
 
===Real numbers===
The absolute value of a [[real number]] ''r'' is defined by:
 
:<math> \left| r \right| = r, \text{ if } r \text{ ≥ } 0 </math>
:<math> \left| r \right| = -r, \text{ if } r < 0 .</math>
 
It may be thought of as the number's distance from zero on the real [[number line]]. For example, the absolute value of both 7 and −7 is 7.
 
===Complex numbers===
A [[complex number]] ''z'' may be viewed as the position of a point ''P'' in a 2-dimensional [[Euclidean space|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]]:
 
:<math> \left| z \right| = \sqrt{a^2 + b^2 }</math>
 
where ''a'' and ''b'' are the respectively [[real part]] and [[imaginary part]] of ''z''. For instance, the modulus of &minus;3 + 4<var>''i''</var> is <math>\sqrt{(-3)^2+4^2} = 5.</math>
 
==Euclidean vectors==
{{Main|Euclidean norm}}
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 number]]s (the [[Cartesian coordinate]]s of ''P''): '''x''' = [''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>]. Its '''magnitude''' or '''length''' is most commonly defined as its [[Norm (mathematics)#Euclidean norm|Euclidean norm]] (or Euclidean length):
:<math>\|\mathbf{x}\| := \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.</math>
For instance, in a 3-dimensional space, the magnitude of [4, 5, 6] is √(4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup>) = √77 or about 8.775.
This is equivalent to the [[square root]] of the [[dot product]] of the vector by itself:
:<math>\|\mathbf{x}\| :=  \sqrt{\mathbf{x} \cdot \mathbf{x}}.</math>
 
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''':
#<math>\left \| \mathbf{x} \right \|,</math>
#<math>\left | \mathbf{x} \right |.</math>
The second notation is generally discouraged{{Weasel-inline|{{subst:2013-10-04}}|date=October 2013}}, because it is also used to denote the [[absolute value]] of scalars and the [[determinant]]s of matrices.
 
==Normed vector spaces==
{{Main|Normed vector space}}
 
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 (mathematics)|norm]]. A [[vector space]] endowed with a norm, such as the Euclidean space, is called a [[normed vector space]]. In high mathematics, not all vector spaces are normed.
 
== Logarithmic magnitudes ==
When comparing magnitudes, it is often helpful to use a [[logarithm]]ic scale. Real-world examples include the [[loudness]] of a [[sound]] ([[decibel]]), the [[brightness]] of a [[star]], or the [[Richter magnitude scale|Richter scale]] of earthquake intensity. Logarithmic magnitudes can be negative. It is usually not meaningful to simply [[addition|add]] or [[subtract]] them.
 
=="Order of magnitude"==
{{main|Order of magnitude}}
 
In advanced mathematics, as well as colloquially in popular culture, especially [[geek culture]], the phrase "order of magnitude" is used to denote a change in a numeric quantity, usually a measurement, by a factor of 10; that is, the moving of the decimal point in a number one way or the other, possibly with the addition of significant zeros.
 
Occasionally the phrase "half an order of magnitude" is also used, generally in more informal contexts. Sometimes, this is used to denote a 5 to 1 change, or alternatively 10<sup>1/2</sup> to 1 (approximately 3.162 to 1).
 
==See also==
*[[Number sense]]
 
==References==
{{reflist}}
 
[[Category:Elementary mathematics]]

Latest revision as of 19:37, 11 January 2015

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