Magnitude (mathematics)

<h2 id="history">History</h2>
<a href="/facts/Ancient_Greeks/PTVBrWXq">Ancient Greeks</a> distinguished between several types of magnitude,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> including:

<ul><li>Positive <a href="/facts/Fraction/ghnQE9B3">fractions</a></li>
<li><a href="/facts/Line_segment/V8E0M4qk">Line segments</a> (ordered by <a href="/facts/Length/EwFyAqGH">length</a>)</li>
<li><a href="/facts/Geometric_shape/nww0x85d">Plane figures</a> (ordered by <a href="/facts/Area/KsNG1G9t">area</a>)</li>
<li><a href="/facts/Solid_geometry/tHH3JFoq">Solids</a> (ordered by <a href="/facts/Volume/aeAXCBUd">volume</a>)</li>
<li><a href="/facts/Angle/gFUTEQYq">Angles</a> (ordered by angular magnitude)</li></ul>
They proved that the first two could not be the same, or even <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> systems of magnitude.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> They did not consider <a href="/facts/Negative_number/LzErzCmI">negative</a> magnitudes to be meaningful, and magnitude is still primarily used in contexts in which <a href="/facts/Zero/4NBRtKHc">zero</a> is either the smallest size or less than all possible sizes.

<h2 id="numbers">Numbers</h2>
Main article: <a href="/facts/Absolute_value/dwJBpEs5">Absolute value</a>
The magnitude of any <a href="/facts/Number/auQSAE2m">number</a> 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is usually called its <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> or modulus, denoted by 
 
 
 
 
 |
 
 x
 
 |
 
 
 
 {\displaystyle |x|}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h3>Real numbers</h3>
The absolute value of a <a href="/facts/Real_number/R02gw5Pb">real number</a> r is defined by:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

|
          r
          |
        
        =
        r
        ,
        
           if 
        
        r
        
           ≥ 
        
        0
      
    
    {\displaystyle \left|r\right|=r,{\text{ if }}r{\text{ ≥ }}0}

|
 r
 |
 
 =
 −
 r
 ,
 
  if 
 
 r
 <
 0.
 
 
 {\displaystyle \left|r\right|=-r,{\text{ if }}r<0.}

Absolute value may also be thought of as the number's <a href="/facts/Distance/hbXlaEFk">distance</a> from zero on the real <a href="/facts/Number_line/gFVyKQHr">number line</a>. For example, the absolute value of both 70 and −70 is 70.

<h3>Complex numbers</h3>
A <a href="/facts/Complex_number/2w7mqI7e">complex number</a> z may be viewed as the position of a point P in a <a href="/facts/Euclidean_space/R2UbzmzM">2-dimensional space</a>, called the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. The absolute value (or <a href="/facts/Modulus_of_complex_number/dwJBpEs5">modulus</a>) 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 <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a> of a vector in a 2-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

|
          z
          |
        
        =
        
          
            
              a
              
                2
              
            
            +
            
              b
              
                2
              
            
          
        
      
    
    {\displaystyle \left|z\right|={\sqrt {a^{2}+b^{2}}}}

where the real numbers a and b are the <a href="/facts/Real_part/2w7mqI7e">real part</a> and the <a href="/facts/Imaginary_part/2w7mqI7e">imaginary part</a> of z, respectively. For instance, the modulus of −3 + 4i is 
 
 
 
 
 
 (
 −
 3
 
 )
 
 2
 
 
 +
 
 4
 
 2
 
 
 
 
 =
 5
 
 
 {\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5}
 
. Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a>, 
 
 
 
 
 
 
 z
 ¯
 
 
 
 
 
 {\displaystyle {\bar {z}}}
 
, where for any complex number 
 
 
 
 z
 =
 a
 +
 b
 i
 
 
 {\displaystyle z=a+bi}
 
, its complex conjugate is 
 
 
 
 
 
 
 z
 ¯
 
 
 
 =
 a
 −
 b
 i
 
 
 {\displaystyle {\bar {z}}=a-bi}
 
.

|
          z
          |
        
        =
        
          
            z
            
              
                
                  z
                  ¯
                
              
            
          
        
        =
        
          
            (
            a
            +
            b
            i
            )
            (
            a
            −
            b
            i
            )
          
        
        =
        
          
            
              a
              
                2
              
            
            −
            a
            b
            i
            +
            a
            b
            i
            −
            
              b
              
                2
              
            
            
              i
              
                2
              
            
          
        
        =
        
          
            
              a
              
                2
              
            
            +
            
              b
              
                2
              
            
          
        
      
    
    {\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}-abi+abi-b^{2}i^{2}}}={\sqrt {a^{2}+b^{2}}}}

(where 
 
 
 
 
 i
 
 2
 
 
 =
 −
 1
 
 
 {\displaystyle i^{2}=-1}
 
).

<h2 id="vector-spaces">Vector spaces</h2>
<h3>Euclidean vector space</h3>
Main article: <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a>
A <a href="/facts/Euclidean_vector/bCLrdksy">Euclidean vector</a> represents the position of a point P in a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. 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 <a href="/facts/Cartesian_coordinate/lkTqvMmB">Cartesian coordinates</a> of P): x = [x1, x2, ..., xn]. Its magnitude or length, denoted by 
 
 
 
 ‖
 x
 ‖
 
 
 {\displaystyle \|x\|}
 
,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> is most commonly defined as its <a href="/facts/Norm_(mathematics)/xIbR4uE1">Euclidean norm</a> (or Euclidean length):<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

‖
        
          x
        
        ‖
        =
        
          
            
              x
              
                1
              
              
                2
              
            
            +
            
              x
              
                2
              
              
                2
              
            
            +
            ⋯
            +
            
              x
              
                n
              
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \|\mathbf {x} \|={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}

For instance, in a 3-dimensional space, the magnitude of [3, 4, 12] is 13 because 
 
 
 
 
 
 
 3
 
 2
 
 
 +
 
 4
 
 2
 
 
 +
 
 12
 
 2
 
 
 
 
 =
 
 
 169
 
 
 =
 13.
 
 
 {\displaystyle {\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.}

This is equivalent to the <a href="/facts/Square_root/AfrzfBdQ">square root</a> of the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of the vector with itself:

‖
        
          x
        
        ‖
        =
        
          
            
              x
            
            ⋅
            
              x
            
          
        
        .
      
    
    {\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}.}

The Euclidean norm of a vector is just a special case of <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a>: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:

<ol><li>
 
 
 
 
 ‖
 
 x
 
 ‖
 
 ,
 
 
 {\displaystyle \left\|\mathbf {x} \right\|,}
 
</li>
<li>
 
 
 
 
 |
 
 x
 
 |
 
 .
 
 
 {\displaystyle \left|\mathbf {x} \right|.}
 
</li></ol>
A disadvantage of the second notation is that it can also be used to denote the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> and the <a href="/facts/Determinant/eGYqJ2TF">determinants</a> of <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a>, which introduces an element of ambiguity.

<h3>Normed vector spaces</h3>
Main article: <a href="/facts/Normed_vector_space/rmDljfyE">Normed vector space</a>
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract <a href="/facts/Vector_space/M1pxMLx2">vector space</a> does not possess a magnitude.
A <a href="/facts/Vector_space/M1pxMLx2">vector space</a> endowed with a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>, such as the Euclidean space, is called a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> The norm of a vector v in a normed vector space can be considered to be the magnitude of v.

<h3>Pseudo-Euclidean space</h3>
In a <a href="/facts/Pseudo-Euclidean_space/UW6sbGeK">pseudo-Euclidean space</a>, the magnitude of a vector is the value of the <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a> for that vector.

<h2 id="logarithmic-magnitudes">Logarithmic magnitudes</h2>
When comparing magnitudes, a <a href="/facts/Logarithmic_scale/CvIGK5TQ">logarithmic scale</a> is often used. Examples include the <a href="/facts/Loudness/N760pyRc">loudness</a> of a <a href="/facts/Sound/DlKXlPuF">sound</a> (measured in <a href="/facts/Decibel/0LyxAPXh">decibels</a>), the <a href="/facts/Brightness/e93R5I4b">brightness</a> of a <a href="/facts/Star/PFbd6ICx">star</a>, and the <a href="/facts/Richter_magnitude_scale/UDhqkAX1">Richter scale</a> of earthquake intensity. Logarithmic magnitudes can be negative. In the <a href="/facts/Natural_sciences/8hg81Q2p">natural sciences</a>, a logarithmic magnitude is typically referred to as a <a href="/facts/Level_(logarithmic_quantity)/0Yt12Pfr">level</a>.

<h2 id="order-of-magnitude">Order of magnitude</h2>
Main article: <a href="/facts/Order_of_magnitude/cWzRP2dM">Order of magnitude</a>
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.

<h2 id="other-mathematical-measures">Other mathematical measures</h2>
This section is an excerpt from <a href="/facts/Measure_(mathematics)/yCq7zlI4">Measure (mathematics)</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Measure_(mathematics)&action=edit">edit</a>]

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the concept of a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> is a generalization and formalization of <a href="/facts/Geometrical_measures/wTQf5ffQ">geometrical measures</a> (<a href="/facts/Length/EwFyAqGH">length</a>, <a href="/facts/Area/KsNG1G9t">area</a>, <a href="/facts/Volume/aeAXCBUd">volume</a>) and other common notions, such as magnitude, <a href="/facts/Mass/HHdc8XmF">mass</a>, and <a href="/facts/Probability/fgYvMhwb">probability</a> of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, <a href="/facts/Integral/V7lyV997">integration theory</a>, and can be generalized to assume <a href="/facts/Signed_measure/vukVf84A">negative values</a>, as with <a href="/facts/Electrical_charge/xxsBfmlB">electrical charge</a>. Far-reaching generalizations (such as <a href="/facts/Spectral_measure/nFN3JjlB">spectral measures</a> and <a href="/facts/Projection-valued_measure/baNsfTzA">projection-valued measures</a>) of measure are widely used in <a href="/facts/Quantum_physics/BRc3Mzgr">quantum physics</a> and physics in general.

The intuition behind this concept dates back to <a href="/facts/Ancient_Greece/PTVBrWXq">ancient Greece</a>, when <a href="/facts/Archimedes/sIeMhSVF">Archimedes</a> tried to calculate the <a href="/facts/Area_of_a_circle/sRUeg8Cl">area of a circle</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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 of <a href="/facts/%25C3%2589mile_Borel/lF9PBa1T">Émile Borel</a>, <a href="/facts/Henri_Lebesgue/ysngVGXc">Henri Lebesgue</a>, <a href="/facts/Nikolai_Luzin/PhrCxp63">Nikolai Luzin</a>, <a href="/facts/Johann_Radon/sJxXdkDo">Johann Radon</a>, <a href="/facts/Constantin_Carath%25C3%25A9odory/ERbINCUr">Constantin Carathéodory</a>, and <a href="/facts/Maurice_Fr%25C3%25A9chet/QXrzxA1s">Maurice Fréchet</a>, among others.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Number_sense/zsgpWf7y">Number sense</a></li>
<li><a href="/facts/Vector_notation/jfavhXKy">Vector notation</a></li>
<li><a href="/facts/Set_size/m6VPojwT">Set size</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Heath, Thomas Smd. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications. <a href="/wiki/T._L._Heath" target="_blank">/wiki/T._L._Heath</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bloch, Ethan D. (2011), The Real Numbers and Real Analysis, Springer, p. 52, ISBN 9780387721774 – via Google Books, The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece. <a href="9780387721774" target="_blank">9780387721774</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Magnitude Definition (Illustrated Mathematics Dictionary)". mathsisfun.com. Retrieved 2020-08-23. <a href="https://www.mathsisfun.com/definitions/magnitude.html" target="_blank">https://www.mathsisfun.com/definitions/magnitude.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2. <a href="978-0-07-148754-2" target="_blank">978-0-07-148754-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha. <a href="https://archive.org/details/complexanalysisi00ahlf" target="_blank">https://archive.org/details/complexanalysisi00ahlf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Nykamp, Duane. "Magnitude of a vector definition". Math Insight. Retrieved August 23, 2020. <a href="https://mathinsight.org/definition/magnitude_vector" target="_blank">https://mathinsight.org/definition/magnitude_vector</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Howard Anton; Chris Rorres (12 April 2010). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-0-470-43205-1 – via Google Books. <a href="978-0-470-43205-1" target="_blank">978-0-470-43205-1</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 978-1-4020-5494-5 <a href="978-1-4020-5494-5" target="_blank">978-1-4020-5494-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Archimedes Measuring the Circle <a href="https://web.archive.org/web/20040703122928/http://www.math.ubc.ca/~cass/archimedes/circle.html" target="_blank">https://web.archive.org/web/20040703122928/http://www.math.ubc.ca/~cass/archimedes/circle.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Heath, T. L. (1897). "Measurement of a Circle". The Works Of Archimedes. Osmania University, Digital Library Of India. Cambridge University Press. pp. 91–98. <a href="http://archive.org/details/worksofarchimede029517mbp" target="_blank">http://archive.org/details/worksofarchimede029517mbp</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Magnitude (mathematics) open-in-new

Magnitude (mathematics)