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Absolute difference

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The absolute difference of two <a href="Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-y|}">

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<annotation encoding="application/x-tex">{\displaystyle |x-y|}</annotation>

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</math><img src="./9eca622011e2b10520e69a7fa83f3bd75159ab66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle |x-y|}" loading="lazy">, the <a href="Absolute_value" title="Absolute value">absolute value</a> of their <a href="Difference_(mathematics)" class="mw-redirect" title="Difference (mathematics)">difference</a>. It describes the distance on the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> between the points corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, and is a special case of the <a href="Lp_space" title="Lp space">Lp distance</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation>

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</math><img src="./b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }" loading="lazy">. Its applications in statistics include the <a href="Absolute_deviation" class="mw-redirect" title="Absolute deviation">absolute deviation</a> from a <a href="Central_tendency" title="Central tendency">central tendency</a>.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Absolute difference has the following properties:

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<annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation>

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</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-0|=x}">

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<annotation encoding="application/x-tex">{\displaystyle |x-0|=x}</annotation>

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</math><img src="./4a8c4b3e23b7fad2c85033faa827ed207ab47b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.054ex; height:2.843ex;" alt="{\displaystyle |x-0|=x}" loading="lazy"> (zero is the <a href="Identity_element" title="Identity element">identity element</a> on non-negative numbers)<a href="#cite_note-taldas-1">[1]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x|=0}">

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<annotation encoding="application/x-tex">{\displaystyle |x-x|=0}</annotation>

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</math><img src="./2673c106f913e2c0597257038e8d5d55a3449323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.054ex; height:2.843ex;" alt="{\displaystyle |x-x|=0}" loading="lazy"> (every element is its own <a href="Inverse_element" title="Inverse element">inverse element</a>)<a href="#cite_note-taldas-1">[1]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle |x-y|\geq 0}</annotation>

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</math><img src="./4974b90acc2830e8b19e594fac18abf6b4690e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.88ex; height:2.843ex;" alt="{\displaystyle |x-y|\geq 0}" loading="lazy"> (non-negativity)<a href="#cite_note-kubrusly-2">[2]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle |x-y|=0}</annotation>

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</math><img src="./5b12a86843a5552fcbe07c401bb0479927aa902a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.88ex; height:2.843ex;" alt="{\displaystyle |x-y|=0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}">

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<annotation encoding="application/x-tex">{\displaystyle x=y}</annotation>

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</math><img src="./409a91214d63eabe46ec10ff3cbba689ab687366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}" loading="lazy"> (nonzero for distinct arguments).<a href="#cite_note-kubrusly-2">[2]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle |x-y|=|y-x|}</annotation>

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</math><img src="./03af5dac8b84deb773f26090627168d637f419f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.337ex; height:2.843ex;" alt="{\displaystyle |x-y|=|y-x|}" loading="lazy"> (<a href="Symmetry_in_mathematics" title="Symmetry in mathematics">symmetry</a> or <a href="Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>).<a href="#cite_note-taldas-1">[1]</a><a href="#cite_note-kubrusly-2">[2]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle |x-z|\leq |x-y|+|y-z|}</annotation>

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</math><img src="./3840b02f13c1abb0b5ffa41ad631e01f0056e954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.488ex; height:2.843ex;" alt="{\displaystyle |x-z|\leq |x-y|+|y-z|}" loading="lazy"> (the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>);<a href="#cite_note-kubrusly-2">[2]</a><a href="#cite_note-3">[3]</a> equality holds if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y\leq z}">

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<annotation encoding="application/x-tex">{\displaystyle x\leq y\leq z}</annotation>

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</math><img src="./2a8611fcfaba012b45d149ab8fb94e2ce390b4c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.343ex;" alt="{\displaystyle x\leq y\leq z}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq y\geq z}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq y\geq z}</annotation>

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</math><img src="./fc3e5803af855223741a8e105e3adc613dde3b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.343ex;" alt="{\displaystyle x\geq y\geq z}" loading="lazy">.</li></ul>

Because it is non-negative, nonzero for distinct arguments, symmetric, and obeys the triangle inequality, the real numbers form a <a href="Metric_space" title="Metric space">metric space</a> with the absolute difference as its distance, the familiar measure of distance along a line.<a href="#cite_note-4">[4]</a> It has been called "the most natural metric space",<a href="#cite_note-FOOTNOTEKhamsiKirk201114-5">[5]</a> and "the most important concrete metric space".<a href="#cite_note-kubrusly-2">[2]</a> This distance generalizes in many different ways to higher dimensions, as a special case of the <a href="Lp_space" title="Lp space">Lp distances</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation>

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</math><img src="./b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }" loading="lazy">, including the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}">

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<annotation encoding="application/x-tex">{\displaystyle p=1}</annotation>

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</math><img src="./c29a2f2fb3f642618036ed7a79712202e7ada924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}">

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<annotation encoding="application/x-tex">{\displaystyle p=2}</annotation>

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</math><img src="./d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy"> cases (<a href="Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a> and <a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a>, respectively). It is also the one-dimensional special case of <a href="Hyperbolic_space" title="Hyperbolic space">hyperbolic distance</a>.

Instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-y|}">

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<annotation encoding="application/x-tex">{\displaystyle |x-y|}</annotation>

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</math><img src="./9eca622011e2b10520e69a7fa83f3bd75159ab66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle |x-y|}" loading="lazy">, the absolute difference may also be expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(x,y)-\min(x,y).}">

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<annotation encoding="application/x-tex">{\displaystyle \max(x,y)-\min(x,y).}</annotation>

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</math> Generalizing this to more than two values, in any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of the real numbers which has an <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a> and a <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a>, the absolute difference between any two numbers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is less or equal then the absolute difference of the infimum and supremum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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The absolute difference takes non-negative integers to non-negative integers. As a binary operation that is commutative but not associative, with an identity element on the non-negative numbers, the absolute difference gives the non-negative numbers (whether real or integer) the algebraic structure of a <a href="Commutative_magma" title="Commutative magma">commutative magma</a> with identity.<a href="#cite_note-taldas-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The absolute difference is used to define the <a href="Relative_difference" class="mw-redirect" title="Relative difference">relative difference</a>, the absolute difference between a given value and a reference value divided by the reference value itself.<a href="#cite_note-6">[6]</a>

In the theory of <a href="Graceful_labeling" title="Graceful labeling">graceful labelings</a> in <a href="Graph_theory" title="Graph theory">graph theory</a>, vertices are labeled by <a href="Natural_number" title="Natural number">natural numbers</a> and edges are labeled by the absolute difference of the numbers at their two vertices. A labeling of this type is graceful when the edge labels are distinct and consecutive from 1 to the number of edges.<a href="#cite_note-7">[7]</a>

As well as being a special case of the Lp distances, absolute difference can be used to define <a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a> (L∞), in which the distance between points is the maximum or supremum of the absolute differences of their coordinates.<a href="#cite_note-8">[8]</a>

In statistics, the <a href="Absolute_deviation" class="mw-redirect" title="Absolute deviation">absolute deviation</a> of a sampled number from a <a href="Central_tendency" title="Central tendency">central tendency</a> is its absolute difference from the center, the <a href="Average_absolute_deviation" title="Average absolute deviation">average absolute deviation</a> is the average of the absolute deviations of a collection of samples, and <a href="Least_absolute_deviations" title="Least absolute deviations">least absolute deviations</a> is a method for <a href="Robust_statistics" title="Robust statistics">robust statistics</a> based on minimizing the average absolute deviation.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFTalukdarDas1996" class="citation journal cs1">Talukdar, D.; Das, N. R. (July 1996). "80.33 Measuring associativity in a groupoid of natural numbers". <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 80 (488): 401–404. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3619592">10.2307/3619592</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3619592">3619592</a>.</cite>

</li>

<li id="cite_note-kubrusly-2">^ <a href="#cite_ref-kubrusly_2-0">a</a> <a href="#cite_ref-kubrusly_2-1">b</a> <a href="#cite_ref-kubrusly_2-2">c</a> <a href="#cite_ref-kubrusly_2-3">d</a> <a href="#cite_ref-kubrusly_2-4">e</a> <cite id="CITEREFKubrusly2001" class="citation book cs1">Kubrusly, Carlos S. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0ijlBwAAQBAJ&pg=PA86">Elements of Operator Theory</a>. Boston: Birkhäuser. p. 86. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-3328-0">10.1007/978-1-4757-3328-0</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781475733280</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFKhamsiKirk2011" class="citation book cs1">Khamsi, Mohamed A.; Kirk, William A. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3ZlpXpedkasC&pg=PA7">"1.3 The triangle inequality in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">"</a>. An Introduction to Metric Spaces and Fixed Point Theory. John Wiley & Sons. pp. 7–8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118031322</bdi>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFGeorgievZennir2019" class="citation book cs1">Georgiev, Svetlin G.; Zennir, Khaled (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1XicDwAAQBAJ&pg=PA25">Functional Analysis with Applications</a>. Walter de Gruyter GmbH. p. 25. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783110657722</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEKhamsiKirk201114-5"><a href="#cite_ref-FOOTNOTEKhamsiKirk201114_5-0">^</a> <a href="#CITEREFKhamsiKirk2011">Khamsi & Kirk (2011)</a>, p. 14.

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFRebaShier2014" class="citation book cs1">Reba, Marilyn A.; Shier, Douglas R. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4rDNBQAAQBAJ&pg=PA463">Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking</a>. CRC Press. p. 463. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781482297935</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFGolomb1972" class="citation book cs1"><a href="Solomon_W._Golomb" title="Solomon W. Golomb">Golomb, Solomon W.</a> (1972). "How to number a graph". In <a href="Ronald_C._Read" title="Ronald C. Read">Read, Ronald C.</a> (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ja7iBQAAQBAJ&pg=PA23">Graph Theory and Computing</a>. Academic Press. pp. 23–37. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB978-1-4832-3187-7.50008-8">10.1016/B978-1-4832-3187-7.50008-8</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0340107">0340107</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFWebb2003" class="citation book cs1">Webb, Andrew R. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ivMBWCe_f0gC&pg=PA421">Statistical Pattern Recognition</a> (2nd ed.). John Wiley & Sons. p. 421. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780470854785</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AbsoluteDifference.html">"Absolute Difference"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>

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</style><div id="Real_numbers16" style="font-size:114%;margin:0 4em"><a href="Real_number" title="Real number">Real numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cantor_set" title="Cantor set">Cantor set</a></li>

<li><a href="Cantor%E2%80%93Dedekind_axiom" title="Cantor–Dedekind axiom">Cantor–Dedekind axiom</a></li>

<li><a href="Completeness_of_the_real_numbers" title="Completeness of the real numbers">Completeness</a></li>

<li><a href="Construction_of_the_real_numbers" title="Construction of the real numbers">Construction</a></li>

<li><a href="Decidability_of_first-order_theories_of_the_real_numbers" title="Decidability of first-order theories of the real numbers">Decidability of first-order theories</a></li>

<li><a href="Extended_real_number_line" title="Extended real number line">Extended real number line</a></li>

<li><a href="Gregory_number" title="Gregory number">Gregory number</a></li>

<li><a href="Irrational_number" title="Irrational number">Irrational number</a></li>

<li><a href="Normal_number" title="Normal number">Normal number</a></li>

<li><a href="Rational_number" title="Rational number">Rational number</a></li>

<li><a href="Rational_zeta_series" title="Rational zeta series">Rational zeta series</a></li>

<li><a href="Real_coordinate_space" title="Real coordinate space">Real coordinate space</a></li>

<li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski axiomatization</a></li>

<li><a href="Vitali_set" title="Vitali set">Vitali set</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-02-26" href="https://en.wikipedia.org/wiki/?title=Absolute_difference&oldid=1277669567">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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