Proportionality (mathematics)

<h2 id="direct-proportionality">Direct proportionality</h2>
See also: <a href="/facts/Equals_sign/AweYSg9h">Equals sign</a>
Given an <a href="/facts/Variable_(mathematics)/VbQlrvKz">independent variable</a> x and a dependent variable y, y is directly proportional to x<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> if there is a positive constant k such that:

y
        =
        k
        x
      
    
    {\displaystyle y=kx}

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter <a href="/facts/Alpha/I51VqVVm">alpha</a>) or "~", with exception of Japanese texts, where "~" is reserved for intervals:

y
 ∝
 x
 
 
 {\displaystyle y\propto x}
 
 (or 
 
 
 
 y
 ∼
 x
 
 
 {\displaystyle y\sim x}
 
)
For 
 
 
 
 x
 ≠
 0
 
 
 {\displaystyle x\neq 0}
 
 the proportionality constant can be expressed as the ratio:

k
        =
        
          
            y
            x
          
        
      
    
    {\displaystyle k={\frac {y}{x}}}

It is also called the constant of variation or constant of proportionality. 
Given such a constant k, the proportionality <a href="/facts/Binary_relation/r9BwGgaK">relation</a> ∝ with proportionality constant k between two sets A and B is the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> defined by 
 
 
 
 {
 (
 a
 ,
 b
 )
 ∈
 A
 ×
 B
 :
 a
 =
 k
 b
 }
 .
 
 
 {\displaystyle \{(a,b)\in A\times B:a=kb\}.}

A direct proportionality can also be viewed as a <a href="/facts/Linear_equation/JfCgALwf">linear equation</a> in two variables with a <a href="/facts/Y-intercept/oJvV25PI">y-intercept</a> of 0 and a <a href="/facts/Slope/a2L3k0j5">slope</a> of k > 0, which corresponds to <a href="/facts/Linear_growth/Ejvy8CKy">linear growth</a>.

<h3>Examples</h3>
<ul><li>If an object travels at a constant <a href="/facts/Speed/I3qgMPfs">speed</a>, then the <a href="/facts/Distance/hbXlaEFk">distance</a> traveled is directly proportional to the <a href="/facts/Time/3ShdpqLx">time</a> spent traveling, with the speed being the constant of proportionality.</li>
<li>The <a href="/facts/Circumference/r1rntU0f">circumference</a> of a <a href="/facts/Circle/9fy5ggKn">circle</a> is directly proportional to its <a href="/facts/Diameter/6uNn4VmU">diameter</a>, with the constant of proportionality equal to <a href="/facts/Pi/vC0UBj1q">π</a>.</li>
<li>On a <a href="/facts/Map/KxQX4B2U">map</a> of a sufficiently small geographical area, drawn to <a href="/facts/Scale_(map)/nw0ytBdk">scale</a> 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.</li>
<li>The <a href="/facts/Force_(physics)/LNAUWMwK">force</a>, acting on a small object with small <a href="/facts/Mass/HHdc8XmF">mass</a> by a nearby large extended mass due to <a href="/facts/Gravity/bqAN7DdQ">gravity</a>, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as <a href="/facts/Gravitational_acceleration/b8IbkkI2">gravitational acceleration</a>.</li>
<li>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, <a href="/facts/Newton%2527s_second_law/XKzGTXK1">Newton's second law</a>, is the classical mass of the object.</li></ul>
<h2 id="inverse-proportionality">Inverse proportionality</h2>

Two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion)<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> if each of the variables is directly proportional to the <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> (reciprocal) of the other, or equivalently if their <a href="/facts/Product_(mathematics)/aXY11nGS">product</a> is a constant.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y
        =
        
          
            k
            x
          
        
      
    
    {\displaystyle y={\frac {k}{x}}}

or equivalently, 
 
 
 
 x
 y
 =
 k
 
 
 {\displaystyle xy=k}
 
. Hence the constant "k" is the product of x and y.
The graph of two variables varying inversely on the <a href="/facts/Cartesian_coordinate/lkTqvMmB">Cartesian coordinate</a> plane is a <a href="/facts/Rectangular_hyperbola/6R6ER0mY">rectangular hyperbola</a>. 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.

<h2 id="hyperbolic-coordinates">Hyperbolic coordinates</h2>
Main article: <a href="/facts/Hyperbolic_coordinates/opVNDavR">Hyperbolic coordinates</a>
The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by <a href="/facts/Hyperbolic_coordinates/opVNDavR">hyperbolic coordinates</a>; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular <a href="/facts/Line_(mathematics)/gJqsr4Gj">ray</a> and the <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a> of inverse proportionality that specifies a point as being on a particular <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a>.

<h2 id="computer-encoding">Computer encoding</h2>
The <a href="/facts/Unicode/DDu3MfPV">Unicode</a> characters for proportionality are the following:

<ul><li>U+221D ∝ PROPORTIONAL TO (&prop;, &Proportional;, &propto;, &varpropto;, &vprop;)</li>
<li>U+007E ~ TILDE</li>
<li>U+2237 ∷ PROPORTION</li>
<li>U+223C ∼ TILDE OPERATOR (&sim;, &thicksim;, &thksim;, &Tilde;)</li>
<li>U+223A ∺ GEOMETRIC PROPORTION (&mDDot;)</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Linear_map/Q1PBKNVV">Linear map</a></li>
<li><a href="/facts/Correlation/egkluAEm">Correlation</a></li>
<li><a href="/facts/Eudoxus_of_Cnidus/FgPAy5L9">Eudoxus of Cnidus</a></li>
<li><a href="/facts/Golden_ratio/anloSRmD">Golden ratio</a></li>
<li><a href="/facts/Inverse-square_law/DxG4T15l">Inverse-square law</a></li>
<li><a href="/facts/Proportional_font/It41hIjl">Proportional font</a></li>
<li><a href="/facts/Ratio/mFmhQt7T">Ratio</a></li>
<li><a href="/facts/Rule_of_three_(mathematics)/MkesbeRU">Rule of three (mathematics)</a></li>
<li><a href="/facts/Sample_size/sf8dAFzq">Sample size</a></li>
<li><a href="/facts/Similarity_(geometry)/Sbd6hEAN">Similarity</a></li>
<li><a href="/facts/Trair%25C4%2581%25C5%259Bika/E34mhbgx">Trairāśika</a></li>
<li><a href="/facts/Basic_proportionality_theorem/zrBsVL2P">Basic proportionality theorem</a></li></ul>
Growth
<ul><li><a href="/facts/Linear_growth/Ejvy8CKy">Linear growth</a></li>
<li><a href="/facts/Hyperbolic_growth/S0Mbef3y">Hyperbolic growth</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Ya. B. Zeldovich, <a href="/facts/Isaak_Yaglom/5f6jeMN4">I. M. Yaglom</a>: Higher math for beginners, <a href="https://books.google.com/books?id=dUB8BwAAQBAJ&pg=PA35">p. 34–35</a>.</li>
<li>Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780877796213, <a href="https://books.google.com/books?id=XeaorGgYAXsC&pg=PA85">p. 85–101</a>.</li>
<li>Lanius, Cynthia S.; Williams Susan E.: <a href="https://www.jstor.org/stable/41181344">PROPORTIONALITY: A Unifying Theme for the Middle Grades</a>. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.</li>
<li>Seeley, Cathy; Schielack Jane F.: <a href="https://www.jstor.org/stable/41182513">A Look at the Development of Ratios, Rates, and Proportionality</a>. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.</li>
<li>Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : <a href="https://www.jstor.org/stable/40539331">Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions</a>. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.</li></ul>

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

<ol>
<li id="fn:1">Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/DirectlyProportional.html" target="_blank">http://mathworld.wolfram.com/DirectlyProportional.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Inverse variation". math.net. Retrieved October 31, 2021. <a href="https://www.math.net/inverse-variation" target="_blank">https://www.math.net/inverse-variation</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/InverselyProportional.html" target="_blank">http://mathworld.wolfram.com/InverselyProportional.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

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Proportionality (mathematics)