View source for Proportionality (mathematics)

{{Short description|Property of two varying quantities with a constant ratio}}
{{Other uses|Proportionality (disambiguation){{!}}Proportionality}}
{{more footnotes needed|date=August 2021}}
[[File:Proportional variables.svg|thumb|300x300px|The variable ''y'' is directly proportional to the variable ''x'' with proportionality constant ~0.6.]]
[[File:Inverse proportionality function plot.gif|thumb|300x300px|The variable ''y'' is inversely proportional to the variable ''x'' with proportionality constant 1.]]

In [[mathematics]], two [[sequence]]s of numbers, often [[experimental data]], are '''proportional''' or '''directly proportional''' if their corresponding elements have a [[Constant (mathematics)|constant]] [[ratio]]. The ratio is called '''''coefficient of proportionality''''' (or '''''proportionality constant''''') and its reciprocal is known as '''''constant of normalization''''' (or '''''normalizing constant'''''). Two sequences are '''inversely proportional''' if corresponding elements have a constant product, also called the coefficient of proportionality.

This definition is commonly extended to related varying quantities, which are often called ''variables''. This meaning of ''variable'' is not the common meaning of the term in mathematics (see [[variable (mathematics)]]); these two different concepts share the same name for historical reasons.

Two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> are ''proportional'' if their ratio <math display=inline>\frac{f(x)}{g(x)}</math> is a [[constant function]].

If several pairs of variables share the same direct proportionality constant, the [[equation]] expressing the equality of these ratios is called a '''proportion''', e.g., {{math|1={{sfrac|''a''|''b''}} = {{sfrac|''x''|''y''}} = ⋯ = ''k''}} (for details see [[Ratio]]).
Proportionality is closely related to ''[[linearity]]''.

== Direct proportionality ==
{{See also|Equals sign}}
Given an [[Variable (mathematics)#Dependent and independent variables|independent variable]] ''x'' and a dependent variable ''y'', ''y'' is '''directly proportional''' to ''x''<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/DirectlyProportional.html "Directly Proportional"]. ''MathWorld'' – A Wolfram Web Resource.</ref> if there is a positive constant ''k'' such that:

: <math>y = kx</math>

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter [[alpha]]) or "~", with exception of Japanese texts, where "~" is reserved for intervals:
: <math>y \propto x</math> (or <math>y \sim  x</math>)

For <math>x \ne 0</math> the '''proportionality constant''' can be expressed as the ratio:

: <math> k = \frac{y}{x}</math>

It is also called the '''constant of variation''' or '''constant of proportionality'''. 
Given such a constant ''k'', the proportionality [[Binary relation|relation]] ∝ with proportionality constant ''k'' between two sets ''A'' and ''B'' is the [[equivalence relation]] defined by <math>\{(a, b) \in A \times B : a = k b\}.</math>

A direct proportionality can also be viewed as a [[linear equation]] in two variables with a [[y-intercept|''y''-intercept]] of {{math|0}} and a [[slope]] of ''k'' > 0, which corresponds to [[linear growth]].

=== Examples ===
* If an object travels at a constant [[speed]], then the [[distance]] traveled is directly proportional to the [[time]] spent traveling, with the speed being the constant of proportionality.
* The [[circumference]] of a [[circle]] is directly proportional to its [[diameter]], with the constant of proportionality equal to [[pi|{{pi}}]].
* On a [[map]] of a sufficiently small geographical area, drawn to [[scale (map)|scale]] distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.
* The [[force (physics)|force]], acting on a small object with small [[mass]] by a nearby large extended mass due to [[gravity]], is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as [[gravitational acceleration]].
* The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, [[Newton's second law]], is the classical mass of the object.

== Inverse proportionality ==
[[File:Inverse proportionality function plot.gif|thumb|300x300px|Inverse proportionality with product {{nowrap|1=''x y'' = 1 .}}]]

Two variables are '''inversely proportional''' (also called '''varying inversely''', in '''inverse variation''', in '''inverse proportion''')<ref>{{cite web | url=https://www.math.net/inverse-variation |title=Inverse variation |website=math.net |access-date=October 31, 2021}}</ref> if each of the variables is directly proportional to the [[multiplicative inverse]] (reciprocal) of the other, or equivalently if their [[Product (mathematics)|product]] is a constant.<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/InverselyProportional.html "Inversely Proportional"]. ''MathWorld'' – A Wolfram Web Resource.</ref> It follows that the variable ''y'' is inversely proportional to the variable ''x'' if there exists a non-zero constant ''k'' such that
: <math>y = \frac{k}{x}</math>
or equivalently, <math>xy = k</math>. Hence the constant "''k''" is the product of ''x'' and ''y''.

The graph of two variables varying inversely on the [[Cartesian coordinate]] plane is a [[rectangular hyperbola]]. The product of the ''x'' and ''y'' values of each point on the curve equals the constant of proportionality (''k''). Since neither ''x'' nor ''y'' can equal zero (because ''k'' is non-zero), the graph never crosses either axis.

Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance (the constant), the time of travel is inversely proportional to speed: ''s''  × ''t'' = ''d''.

== Hyperbolic coordinates ==
{{Main|Hyperbolic coordinates}}
The concepts of ''direct'' and ''inverse'' proportion lead to the location of points in the Cartesian plane by [[hyperbolic coordinates]]; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular [[line (mathematics)#Ray|ray]] and the [[constant (mathematics)|constant]] of inverse proportionality that specifies a point as being on a particular [[hyperbola]].

== Computer encoding ==

The [[Unicode]] characters for proportionality are the following:
* {{unichar|221d|proportional to|html=}}
* {{unichar|7e|tilde|html=}}
* {{Unichar|2237|PROPORTION}}
* {{unichar|223c|tilde operator|html=}}
* {{unichar|223a|GEOMETRIC PROPORTION|html=}}

== See also ==
* [[Linear map]]
* [[Correlation]]
* [[Eudoxus of Cnidus]]
* [[Golden ratio]]
* [[Inverse-square law]]
* [[Proportional font]]
* [[Ratio]]
* [[Rule of three (mathematics)]]
* [[Sample size]]
* [[Similarity (geometry)|Similarity]]
* [[Trairāśika]]
* [[Basic proportionality theorem]]
;Growth 
* [[Linear growth]]
* [[Hyperbolic growth]]

== Notes ==
{{Reflist}}

== References ==
* Ya. B. Zeldovich,  [[Isaak Yaglom|I. M. Yaglom]]: ''Higher math for beginners'', [https://books.google.com/books?id=dUB8BwAAQBAJ&pg=PA35 p. 34–35].
* Brian Burrell: ''Merriam-Webster's Guide to Everyday Math: A Home and Business Reference''. Merriam-Webster, 1998, {{isbn|9780877796213}}, [https://books.google.com/books?id=XeaorGgYAXsC&pg=PA85 p. 85–101].
* Lanius, Cynthia S.; Williams Susan E.: [https://www.jstor.org/stable/41181344 ''PROPORTIONALITY: A Unifying Theme for the Middle Grades'']. Mathematics Teaching in the Middle School 8.8 (2003), p.&nbsp;392–396.
* Seeley, Cathy; Schielack Jane F.: [https://www.jstor.org/stable/41182513 ''A Look at the Development of Ratios, Rates, and Proportionality'']. Mathematics Teaching in the Middle School, 13.3, 2007, p.&nbsp;140–142.
* Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : [https://www.jstor.org/stable/40539331 ''Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions'']. Journal for Research in Mathematics Education, 40.2, 2009, p.&nbsp;187–211.

{{Authority control}}

{{DEFAULTSORT:Proportionality Mathematics}}
[[Category:Mathematical terminology]]
[[Category:Ratios]]