@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: a skos:ConceptScheme .
psr:-WX8H0134-J
  skos:prefLabel "nombre irrationnel"@fr, "irrational number"@en ;
  a skos:Concept ;
  skos:narrower psr:-SDKXNB3P-N .

psr:-SDKXNB3P-N
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:altLabel "quadratic irrational number"@en, "quadratic surd"@en ;
  skos:definition """Un <b>irrationnel quadratique</b> est un nombre irrationnel solution d'une équation quadratique à coefficients rationnels, autrement dit, un nombre réel algébrique de degré 2. Il engendre donc un corps quadratique réel ℚ(<span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"><i>d</i></span></span>), où <span class="texhtml"><i>d</i></span> est un entier positif sans facteur carré. Les irrationnels quadratiques sont caractérisés par la périodicité à partir d'un certain rang de leur développement en fraction continue (théorème de Lagrange). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Irrationnel_quadratique">https://fr.wikipedia.org/wiki/Irrationnel_quadratique</a>)"""@fr, """In mathematics, a <b>quadratic irrational number</b> (also known as a <b>quadratic irrational</b> or <b>quadratic surd</b>) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers.  Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as  <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{\\\\sqrt {c}} \\\\over d},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>a</mi>               <mo>+</mo>               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <msqrt>                   <mi>c</mi>                 </msqrt>               </mrow>             </mrow>             <mi>d</mi>           </mfrac>         </mrow>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {a+b{\\\\sqrt {c}} \\\\over d},}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab827ce7aa5f643c200a8df6dfb38187361d3113" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.493ex; height:6.009ex;" alt="{\\\\displaystyle {a+b{\\\\sqrt {c}} \\\\over d},}"></span></dd></dl> for integers <span class="texhtml"><i>a</i>, <i>b</i>, <i>c</i>, <i>d</i></span>; with <span class="texhtml"><i>b</i></span>, <span class="texhtml"><i>c</i></span> and <span class="texhtml"><i>d</i></span> non-zero, and with <span class="texhtml"><i>c</i></span> square-free. When <span class="texhtml"><i>c</i></span> is positive, we get <b>real quadratic irrational numbers</b>, while a negative <span class="texhtml"><i>c</i></span> gives <b>complex quadratic irrational numbers</b> which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Quadratic_irrational_number">https://en.wikipedia.org/wiki/Quadratic_irrational_number</a>)"""@en ;
  skos:prefLabel "quadratic irrational"@en, "irrationnel quadratique"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Irrationnel_quadratique>, <https://en.wikipedia.org/wiki/Quadratic_irrational_number> ;
  skos:related psr:-LK8XNN3X-R ;
  skos:broader psr:-WX8H0134-J .

psr:-LK8XNN3X-R
  skos:prefLabel "équation du second degré"@fr, "quadratic equation"@en ;
  a skos:Concept ;
  skos:related psr:-SDKXNB3P-N .