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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 ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.

In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude can be defined as quantity or distance. An order of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.

History

Ancient Greeks distinguished between several types of magnitude,^[1] including:

Positive fractions
Line segments (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.^[2] 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

The magnitude of any number $x$ is usually called its absolute value or modulus, denoted by $|x|$ .^[3]

Real numbers

The absolute value of a real number r is defined by:^[4]

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

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

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 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 z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:^[5]

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

where the real numbers a and b are the real part and the imaginary part of z, respectively. For instance, the modulus of −3 + 4i is ${\sqrt {(-3)^{2}+4^{2}}}=5$ . Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, ${\bar {z}}$ , where for any complex number $z=a+bi$ , its complex conjugate is ${\bar {z}}=a-bi$ .

\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 $i^{2}=-1$ ).

Vector spaces

Euclidean vector space

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 coordinates of P): x = [x₁, x₂, ..., x_n]. Its magnitude or length, denoted by $\|x\|$ ,^[6] is most commonly defined as its Euclidean norm (or Euclidean length):^[7]

\|\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 ${\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.$ This is equivalent to the square root of the dot product of the vector with itself:

\|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}.

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:

$\left\|\mathbf {x} \right\|,$
$\left|\mathbf {x} \right|.$

A disadvantage of the second notation is that it can also be used to denote the absolute value of scalars and the determinants of matrices, which introduces an element of ambiguity.

Normed vector spaces

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, such as the Euclidean space, is called a normed vector space.^[8] 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 logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. In the natural sciences, a logarithmic magnitude is typically referred to as a level.

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.

Other mathematical measures

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle.^[9]^[10] 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 Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.

References

^ 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.
^ 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.
^ "Magnitude Definition (Illustrated Mathematics Dictionary)". mathsisfun.com. Retrieved 2020-08-23.
^ Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2.
^ Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha.
^ Nykamp, Duane. "Magnitude of a vector definition". Math Insight. Retrieved August 23, 2020.
^ Howard Anton; Chris Rorres (12 April 2010). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-0-470-43205-1 – via Google Books.
^ Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 978-1-4020-5494-5
^ Archimedes Measuring the Circle
^ Heath, T. L. (1897). "Measurement of a Circle". The Works Of Archimedes. Osmania University, Digital Library Of India. Cambridge University Press. pp. 91–98.

[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.

[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.

[3] "Magnitude Definition (Illustrated Mathematics Dictionary)". mathsisfun.com. Retrieved 2020-08-23.

[4] Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2.

[5] Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha.

[6] Nykamp, Duane. "Magnitude of a vector definition". Math Insight. Retrieved August 23, 2020.

[AntonRorres2010-7] Howard Anton; Chris Rorres (12 April 2010). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-0-470-43205-1 – via Google Books.

[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

[9] Archimedes Measuring the Circle

[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.

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@@ Line 1: / Line 1: @@
+{{Short description|Property determining comparison and ordering}}
-== Real numbers ==
+{{Other uses|Magnitude (disambiguation)}}
-The magnitude of a real number is usually called the '''[[absolute value]]''' or '''modulus'''. It is written | ''x'' |, and is defined by:
+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. Magnitude as a concept dates to [[Ancient Greece]] and has been applied as a [[Measure (mathematics)|measure]] of distance from one object to another. For numbers, the [[absolute value]] of a number is commonly applied as the measure of units between a number and zero.
-: | ''x'' | = ''x'' , if ''x'' &ge; 0
-: | ''x'' | = -''x'' , if ''x'' < 0
+In vector spaces, the [[Euclidean norm]] is a measure of magnitude used to define a distance between two points in space. In [[physics]], magnitude can be defined as quantity or distance. An [[order of magnitude]] is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.
-This gives the number's distance from zero on the real [[number line]]. For example, the modulus of -5 is 5.
-== Complex numbers ==
+== History ==
+[[Ancient Greeks]] distinguished between several types of magnitude,<ref>{{Cite book |last=Heath |first=Thomas Smd. |url=https://archive.org/details/thirteenbooksofe00eucl |title=The Thirteen Books of Euclid's Elements |publisher=Dover Publications |year=1956 |edition=2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] |location=New York |author-link=T. L. Heath |url-access=registration}}</ref> including:
-Similarly, the magnitude of a [[complex number]], called the '''modulus''', gives the distance from zero in the [[Argand diagram]]. The formula for the modulus is the same as that for [[Pythagoras' theorem]].
+* Positive [[fraction]]s
+* [[Line segment]]s (ordered by [[length]])
+* [[Geometric shape|Plane figures]] (ordered by [[area]])
+* [[Solid geometry|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 |last=Bloch |first=Ethan D. |title=The Real Numbers and Real Analysis |url=https://books.google.com/books?id=vXw_AAAAQBAJ&pg=PA52 |page=52 |year=2011 |publisher=Springer |isbn=9780387721774 |quote=The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece. |via=[[Google Books]]}}</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.
-: | ''x'' + ''iy'' | = &radic; ( ''x''&sup2 + ''y''&sup2 )
+== Numbers ==
-For instance, the modulus of -3 + 4<var>i</var> is 5.
+{{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>{{Cite web |title=Magnitude Definition (Illustrated Mathematics Dictionary) |url=https://www.mathsisfun.com/definitions/magnitude.html |access-date=2020-08-23 |website=mathsisfun.com}}</ref>
-== Euclidean vectors ==
-The magnitude of a [[vector (spatial)|vector]] of real numbers in a [[Euclidean space|Euclidean <var>n</var>-space]] is most often the Euclidean norm, derived from [[Euclidean distance]]: the [[square root]] of the [[dot product]] of the vector with itself:
+=== Real numbers ===
-:<math>s=\sqrt{u^2+v^2+w^2}</math>
+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 |date=2008 |publisher=McGraw-Hill Professional |isbn=978-0-07-148754-2 |page=2}}</ref>
-where ''u'', ''v'' and ''w'' are the components.
+: <math> \left| r \right| = r, \text{ if } r \text{ ≥ } 0 </math>
-For instance, the magnitude of [4, 5, 6] is &radic;(4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup>) = &radic;77 or about 8.775.
+: <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.
-== General vector spaces ==
-A concept of length can be applied to a vector space in general. This is then called a [[normed vector space]]. The function that maps objects to their magnitudes is called a [[norm]].
+=== 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 complex number|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. |url=https://archive.org/details/complexanalysisi00ahlf |title=Complex Analysis |date=1953 |publisher=McGraw Hill Kogakusha |location=Tokyo |url-access=registration}}</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>, where for any complex number <math> z = a + bi</math>, its complex conjugate is <math> \bar{z} = a -bi</math>.
+:<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>
+(where <math>i^2 = -1</math>).
+== 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 numbers (the [[Cartesian coordinate]]s of ''P''): ''x'' = [''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>]. Its '''magnitude''' or '''length''', denoted by <math>\|x\|</math>,<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 |last=Howard Anton |url=https://books.google.com/books?id=1PJ-WHepeBsC&q=magnitude |title=Elementary Linear Algebra: Applications Version |last2=Chris Rorres |date=12 April 2010 |publisher=John Wiley & Sons |isbn=978-0-470-43205-1 |via=[[Google Books]]}}</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,&nbsp;4,&thinsp;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 with 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 [[matrix (mathematics)|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. |title=The Linear Algebra a Beginning Graduate Student Ought to Know |date=January 2007 |edition=2nd |publisher=Springer |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 [[logarithmic 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.  In the [[natural sciences]], a logarithmic magnitude is typically referred to as a ''[[Level (logarithmic quantity)|level]]''.
+== 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.
+== Other mathematical measures ==
+{{excerpt|Measure (mathematics)}}
+== See also ==
+* [[Number sense]]
+* [[Vector notation]]
+* [[Set size]]
+== References ==
+{{Reflist}}
+[[Category:Elementary mathematics]]
+[[Category:Unary operations]]