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'{{Other uses|Magnitude (disambiguation)}} 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 the displayed result of 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,<ref>{{cite book | last = Heath | first = Thomas Smd. | authorlink = T. L. Heath | title = The Thirteen Books of Euclid's Elements | 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 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]] is usually called its "[[absolute value]]" or "modulus", denoted by |''x''|. ===Real numbers=== The absolute value of a [[real number]] ''r'' is defined by:<ref>Mendelson, Elliott, ''Schaum's Outline of Beginning Calculus'', McGraw-Hill Professional, 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> 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 [[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>Ahlfors, Lars V.: ''Complex Analysis'', Mc Graw Hill Kogakusha, Tokyo (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]], ''z''^∗, where for any complex number {{nowrap|1=''z'' = ''a'' + ''bi''}}, its complex conjugate is {{nowrap|1=''z''^∗ = ''a'' − ''bi''}}. :<math> \left| z \right| = \sqrt{zz^* } = \sqrt{(a+bi)(a-bi)} = \sqrt{a^2 -abi + abi - b^2i^2} = \sqrt{a^2 + b^2 }</math> <big>(</big> recall <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''' is most commonly defined as its [[Norm (mathematics)#Euclidean norm|Euclidean norm]] (or Euclidean length):<ref> {{Citation|last=Anton|first=Howard|year=2005|title=Elementary Linear Algebra (Applications Version)|publisher=Wiley International|edition=9th}}</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 √(3² + 4² + 12²) = √169 = 13. 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 to the second notation is that it is also used to denote the [[absolute value]] of scalars and the [[determinant]]s of matrices and therefore its meaning can be ambiguous. ===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]].<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> Not all vector spaces are normed. ===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. It is not meaningful to simply add or subtract them. ==Order of magnitude== {{main|Order of magnitude}} The phrase "an order of magnitude different" is used to denote a difference in a numeric quantity, usually a measurement, by a factor of 10—that is, a difference of one digit in the location of the decimal point. ==See also== *[[Number sense]] ==References== {{reflist}} [[Category:Elementary mathematics]]'