Edit filter log

'{{Other uses|Magnitude (disambiguation)}} In mathematics, '''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. In physics, Magnitude poopy pants can be defined as quantity or distance ==History== The Greeks distinguished between several types of magnitude,<ref>{{cite book | last = Heath | first = Thomas Smd. | author-link = T. L. Heath | title = The Thirteen Books of Euclid's Elements | url = https://archive.org/details/thirteenbooksofe00eucl | url-access = registration | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] | year = 1956 | publisher = Dover Publications | location = New York }}</ref> including: *Positive [[fractions]] *[[Line segment]]s (ordered by [[length]]) *[[Geometric shape|Plane figures]] (ordered by [[area]]) *[[Polyhedron|Solids]] (ordered by [[volume]]) *[[Angle|Angles]] (ordered by angular magnitude) 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 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. ==Numbers== {{Main|Absolute value}} 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> ===Real numbers=== 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> :<math> \left| r \right| = r, \text{ if } r \text{ ≥ } 0 </math> :<math> \left| r \right| = -r, \text{ if } r < 0 .</math> 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. ===Complex numbers=== 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> :<math> \left| z \right| = \sqrt{a^2 + b^2 }</math> 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 {{nowrap|1=''z'' = ''a'' + ''bi''}}, its complex conjugate is {{nowrap|1=''z''^∗ = ''a'' − ''bi''}}. :<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> <big>(</big>where <math>i^2 = -1</math><big>)</big> ==Vector spaces== ===Euclidean vector space=== {{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''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> :<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 [3, 4, 12] is 13 because <math>\sqrt{3^2 + 4^2 + 12^2} = \sqrt{169} = 13.</math> 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> 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 matrices, which introduces an element of ambiguity. ===Normed vector spaces=== {{Main|Normed vector space}} By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. 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''. ===Pseudo-Euclidean space=== In a [[pseudo-Euclidean space]], the magnitude of a vector is the value of the [[quadratic form]] for that vector. ==Logarithmic magnitudes== 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). ==Order of magnitude== {{main|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 sense]] *[[Vector notation]] ==References== {{reflist}} [[Category:Elementary mathematics]] [[Category:Unary operations]]'