@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-LLND57KL-D
  skos:prefLabel "algèbre associative"@fr, "associative algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-RWVJX5RR-G .

psr:-RXQC777M-K
  skos:prefLabel "algèbre différentielle"@fr, "differential algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-RWVJX5RR-G .

psr: a skos:ConceptScheme .
psr:-RWVJX5RR-G
  a skos:Concept ;
  skos:definition """In algebra, the <b>dual numbers</b> are a hypercomplex number system first introduced in the 19th century. They are expressions of the form <span class="texhtml"><i>a</i> + <i>bε</i></span>, where <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are real numbers, and <span class="texhtml mvar" style="font-style:italic;">ε</span> is a symbol taken to satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varepsilon ^{2}=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>ε<!-- ε --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varepsilon ^{2}=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68eddcca75ba10af115dde98b267c3afd5341d40" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.399ex; height:2.676ex;" alt="{\\\\displaystyle \\\\varepsilon ^{2}=0}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varepsilon \\
eq 0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ε<!-- ε --></mi>
<br/>        <mo>≠<!-- ≠ --></mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varepsilon \\
eq 0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b164106e88d10861e8dbc94194c20a96202efbde" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.676ex;" alt="{\\\\displaystyle \\\\varepsilon \\
eq 0}"></span>.
<br/>Dual numbers can be added component-wise, and multiplied by the formula
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (a+b\\\\varepsilon )(c+d\\\\varepsilon )=ac+(ad+bc)\\\\varepsilon ,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>+</mo>
<br/>        <mi>b</mi>
<br/>        <mi>ε<!-- ε --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>c</mi>
<br/>        <mo>+</mo>
<br/>        <mi>d</mi>
<br/>        <mi>ε<!-- ε --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>a</mi>
<br/>        <mi>c</mi>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mi>d</mi>
<br/>        <mo>+</mo>
<br/>        <mi>b</mi>
<br/>        <mi>c</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>ε<!-- ε --></mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a+b\\\\varepsilon )(c+d\\\\varepsilon )=ac+(ad+bc)\\\\varepsilon ,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929f0e02bb2164de3e36b32b990a12a25637bae4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:34.922ex; height:2.843ex;" alt="{\\\\displaystyle (a+b\\\\varepsilon )(c+d\\\\varepsilon )=ac+(ad+bc)\\\\varepsilon ,}"></span></dd></dl>
<br/>which follows from the property <span class="texhtml"><i>ε</i><sup>2</sup> = 0</span> and the fact that multiplication is a bilinear operation.
<br/>The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Dual_number">https://en.wikipedia.org/wiki/Dual_number</a>)"""@en, """En mathématiques et en algèbre abstraite, les <b>nombres duaux</b> sont une algèbre associative unitaire commutative à deux dimensions sur les nombres réels, apparaissant à partir des réels par adjonction d'un nouvel élément ε avec la propriété ε<sup>2</sup> = 0 (ε est un élément nilpotent). Ils ont été introduits par William Clifford en 1873.
<br/>Ils sont notamment utiles pour fournir un outil de dérivation automatique. Ils ont également des applications en physique. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_dual">https://fr.wikipedia.org/wiki/Nombre_dual</a>)"""@fr ;
  dc:created "2023-08-23"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Dual_number>, <https://fr.wikipedia.org/wiki/Nombre_dual> ;
  skos:prefLabel "nombre dual"@fr, "dual number"@en ;
  skos:broader psr:-FTGGBTC5-X, psr:-LLND57KL-D, psr:-X4R4FW27-X, psr:-RXQC777M-K ;
  dc:modified "2023-09-22"^^xsd:date ;
  skos:inScheme psr: .

psr:-X4R4FW27-X
  skos:prefLabel "hypercomplex number"@en, "nombre hypercomplexe"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RWVJX5RR-G .

psr:-FTGGBTC5-X
  skos:prefLabel "algèbre commutative"@fr, "commutative algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-RWVJX5RR-G .