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[ 0 => '{{Other uses|Magnitude (disambiguation)}}', 1 => '', 2 => 'In [[mathematics]], the '''magnitude''' or '''size''' of a [[mathematical 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 an [[order theory|ordering]] (or ranking)—of the [[class (mathematics)|class]] of objects to which it belongs.', 3 => '', 4 => 'In [[physics]], magnitude can be defined as quantity or distance.', 5 => '', 6 => '==History==', 7 => 'The Greeks distinguished between several types of magnitude,<ref>{{cite book', 8 => ' | last = Heath', 9 => ' | first = Thomas Smd.', 10 => ' | author-link = T. L. Heath', 11 => ' | title = The Thirteen Books of Euclid's Elements', 12 => ' | url = https://archive.org/details/thirteenbooksofe00eucl', 13 => ' | url-access = registration', 14 => ' | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925]', 15 => ' | year = 1956', 16 => ' | publisher = Dover Publications', 17 => ' | location = New York', 18 => ' }}</ref> including:', 19 => '*Positive [[fraction]]s', 20 => '*[[Line segment]]s (ordered by [[length]])', 21 => '*[[Geometric shape|Plane figures]] (ordered by [[area]])', 22 => '*[[Solid geometry|Solids]] (ordered by [[volume]])', 23 => '*[[Angle|Angles]] (ordered by angular magnitude)', 24 => '', 25 => 'They proved that the first two could not be the same, or even [[isomorphic]] systems of magnitude.<ref>{{citation|title=The Real Numbers and Real Analysis|first=Ethan D.|last=Bloch|publisher=Springer|year=2011|isbn=9780387721774|page=52|url=https://books.google.com/books?id=vXw_AAAAQBAJ&pg=PA52|quote=The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece}}.</ref> They did not consider [[negative number|negative]] magnitudes to be meaningful, and ''magnitude'' is still primarily used in contexts in which [[zero]] is either the smallest size or less than all possible sizes.', 26 => '', 27 => '==Numbers==', 28 => '{{Main|Absolute value}}', 29 => '', 30 => 'The magnitude of any [[number]] <math>x</math> is usually called its ''[[absolute value]]'' or ''modulus'', denoted by <math>|x|</math>.<ref name=":0">{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-08-23|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|title=Magnitude Definition (Illustrated Mathematics Dictionary)|url=https://www.mathsisfun.com/definitions/magnitude.html|access-date=2020-08-23|website=www.mathsisfun.com}}</ref>', 31 => '', 32 => '===Real numbers===', 33 => 'The absolute value of a [[real number]] ''r'' is defined by:<ref>{{cite book|last=Mendelson|first=Elliott|title=Schaum's Outline of Beginning Calculus|publisher=McGraw-Hill Professional|date=2008|isbn=978-0-07-148754-2|page=2}}</ref>', 34 => '', 35 => ':<math> \left| r \right| = r, \text{ if } r \text{ ≥ } 0 </math>', 36 => ':<math> \left| r \right| = -r, \text{ if } r < 0 .</math>', 37 => '', 38 => 'Absolute value may also 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.', 39 => '', 40 => '===Complex numbers===', 41 => 'A [[complex number]] ''z'' may be viewed as the position of a point ''P'' in a [[Euclidean space|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 {{nowrap|1=''z'' = ''a'' + ''bi''}} is similar to that for the [[Euclidean norm]] of a vector in a 2-dimensional [[Euclidean space]]:<ref>{{cite book|last=Ahlfors|first=Lars V.|title=Complex Analysis|url=https://archive.org/details/complexanalysisi00ahlf|url-access=registration|publisher=McGraw Hill Kogakusha|location=Tokyo|date=1953}}</ref>', 42 => '', 43 => ':<math>\left| z \right| = \sqrt{a^2 + b^2}</math>', 44 => '', 45 => 'where the real numbers ''a'' and ''b'' are the [[real part]] and the [[imaginary part]] of ''z'', respectively. For instance, the modulus of {{nowrap|−3 + 4<var>''i''</var>}} is <math>\sqrt{(-3)^2+4^2} = 5</math>. Alternatively, the magnitude of a complex number ''z'' may be defined as the square root of the product of itself and its [[complex conjugate]], <math>\bar{z}</math>,<ref name=":0" /> where for any complex number <math> z = a + bi</math>, its complex conjugate is <math> \bar{z} = a -bi</math>. ', 46 => ':<math> \left| z \right| = \sqrt{z\bar{z} } = \sqrt{(a+bi)(a-bi)} = \sqrt{a^2 -abi + abi - b^2i^2} = \sqrt{a^2 + b^2 }</math>', 47 => '(where <math>i^2 = -1</math>).', 48 => '', 49 => '==Vector spaces==', 50 => '===Euclidean vector space===', 51 => '{{Main|Euclidean norm}}', 52 => '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 coordinate]]s of ''P''): ''x'' = [''x''₁, ''x''₂, ..., ''x''_''n'']. Its '''magnitude''' or '''length''', denoted by <math>\|x\|</math>,<ref name=":0" /><ref>{{Cite web|last=Nykamp|first=Duane|title=Magnitude of a vector definition|url=https://mathinsight.org/definition/magnitude_vector|access-date=August 23, 2020|website=Math Insight}}</ref> is most commonly defined as its [[Norm (mathematics)#Euclidean norm|Euclidean norm]] (or Euclidean length):<ref name="AntonRorres2010">{{cite book|author1=Howard Anton|author2=Chris Rorres|title=Elementary Linear Algebra: Applications Version|url=https://books.google.com/books?id=1PJ-WHepeBsC&q=magnitude|date=12 April 2010|publisher=John Wiley & Sons|isbn=978-0-470-43205-1}}</ref>', 53 => ':<math>\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.</math>', 54 => 'For instance, in a 3-dimensional space, the magnitude of [3,&nbsp;4,&thinsp;12] is 13 because <math>\sqrt{3^2 + 4^2 + 12^2} = \sqrt{169} = 13.</math>', 55 => 'This is equivalent to the [[square root]] of the [[dot product]] of the vector with itself:', 56 => ':<math>\|\mathbf{x}\| = \sqrt{\mathbf{x} \cdot \mathbf{x}}.</math>', 57 => '', 58 => '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'':', 59 => '#<math>\left \| \mathbf{x} \right \|,</math>', 60 => '#<math>\left | \mathbf{x} \right |.</math>', 61 => 'A disadvantage of the second notation is that it can also be used to denote the [[absolute value]] of [[Scalar (mathematics)|scalars]] and the [[determinant]]s of [[matrix (mathematics)|matrices]], which introduces an element of ambiguity.', 62 => '', 63 => '===Normed vector spaces===', 64 => '{{Main|Normed vector space}}', 65 => '', 66 => 'By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract [[vector space]] does not possess a magnitude.', 67 => '', 68 => 'A [[vector space]] endowed with a [[norm (mathematics)|norm]], such as the Euclidean space, is called a [[normed vector space]].<ref>{{Citation|last=Golan|first=Johnathan S.|date=January 2007|title=The Linear Algebra a Beginning Graduate Student Ought to Know|publisher=Springer|edition=2nd|isbn=978-1-4020-5494-5}}</ref> The norm of a vector ''v'' in a normed vector space can be considered to be the magnitude of ''v''.', 69 => '', 70 => '===Pseudo-Euclidean space===', 71 => 'In a [[pseudo-Euclidean space]], the magnitude of a vector is the value of the [[quadratic form]] for that vector.', 72 => '', 73 => '==Logarithmic magnitudes==', 74 => 'When comparing magnitudes, a [[logarithm]]ic scale is often used. Examples include the [[loudness]] of a [[sound]] (measured in [[decibel|decibels]]), the [[brightness]] of a [[star]], and the [[Richter magnitude scale|Richter scale]] of earthquake intensity. Logarithmic magnitudes can be negative, and cannot be added or subtracted meaningfully (since the relationship is non-linear).', 75 => '', 76 => '==Order of magnitude==', 77 => '{{main|Order of magnitude}}', 78 => '', 79 => '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.', 80 => '', 81 => '==See also==', 82 => '*[[Number sense]]', 83 => '*[[Vector notation]]', 84 => '', 85 => '==References==', 86 => '{{reflist}}', 87 => '', 88 => '[[Category:Elementary mathematics]]', 89 => '[[Category:Unary operations]]' ]