Editing Magnitude (mathematics)

{{Short description|Property determining comparison and ordering}}
{{Other uses|Magnitude (disambiguation)}}

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.

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,<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:
* 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.

== 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>{{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>

=== 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 |date=2008 |publisher=McGraw-Hill Professional |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 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]]