Jacques Philippe Marie Binet

<h2 id="career">Career</h2>
<p>Binet graduated from l'<a href="/facts/%25C3%2589cole_Polytechnique/0DS7aMaR">École Polytechnique</a> in 1806, and returned as a teacher in 1807. He advanced in position until 1816 when he became an inspector of studies at l'École. He held this post until 13 November 1830, when he was dismissed by the recently sworn in <a href="/facts/King_Louis-Philippe/DMuYlVpC">King Louis-Philippe</a> of France, probably because of Binet's strong support of the previous King, <a href="/facts/Charles_X_of_France/9W515c3S">Charles X</a>. In 1823 Binet succeeded <a href="/facts/Delambre/DEBdp8Rj">Delambre</a> in the chair of <a href="/facts/Astronomy/0ArJVM1p">astronomy</a> at the <i><a href="/facts/Coll%25C3%25A8ge_de_France/zDuHr8f1">Collège de France</a></i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> He was made a <a href="/facts/Knight/q6oIppem">Chevalier</a> in the <a href="/facts/L%25C3%25A9gion_d%2527Honneur/XmLQTz71">Légion d'Honneur</a> in 1821, and was elected to the <a href="/facts/Acad%25C3%25A9mie_des_Sciences/zF42DYQv">Académie des Sciences</a> in 1843.
</p>
<h2 id="binets-fibonacci-number-formula">Binet's Fibonacci number formula</h2>
<p class="note">Main article: <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci number § Closed-form expression</a></p>
<p>The <a href="/facts/Fibonacci_sequence/mqxpAUQD">Fibonacci sequence</a> is defined by
</p>
<ul><li>
  
    
      
        
          u
          
            0
          
        
        =
        0
      
    
    {\displaystyle u_{0}=0}
  
</li>
<li>
  
    
      
        
          u
          
            1
          
        
        =
        1
      
    
    {\displaystyle u_{1}=1}
  
</li>
<li>
  
    
      
        
          u
          
            n
          
        
        =
        
          u
          
            n
            −
            1
          
        
        +
        
          u
          
            n
            −
            2
          
        
        ,
        
           for 
        
        n
        >
        1
      
    
    {\displaystyle u_{n}=u_{n-1}+u_{n-2},{\text{ for }}n>1}
  
</li></ul>
<p>Binet's formula provides a <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form expression</a> for the 
  
    
      
        
          n
          
            th
          
        
      
    
    {\displaystyle n^{\text{th}}}
  
 term in this sequence:
</p>

u
          
            n
          
        
        =
        
          
            
              (
              1
              +
              
                
                  5
                
              
              
                )
                
                  n
                
              
              −
              (
              1
              −
              
                
                  5
                
              
              
                )
                
                  n
                
              
            
            
              
                2
                
                  n
                
              
              
                
                  5
                
              
            
          
        
      
    
    {\displaystyle u_{n}={\frac {(1+{\sqrt {5}})^{n}-(1-{\sqrt {5}})^{n}}{2^{n}{\sqrt {5}}}}}
  
      <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
<p>Given:
</p><p>
  
    
      
        φ
        =
        
          
            
              (
              1
              +
              
                
                  5
                
              
              )
            
            2
          
        
      
    
    {\displaystyle \varphi ={(1+{\sqrt {5}}) \over 2}}

</p><p>a simplified version of Binet's formula is:
</p><p>
  
    
      
        
          u
          
            n
          
        
        =
        
          ⌊
          
            
              
                
                  φ
                  
                    n
                  
                
                
                  
                    5
                  
                
              
            
            +
            
              
                1
                2
              
            
          
          ⌋
        
      
    
    {\displaystyle u_{n}=\left\lfloor {{\varphi ^{n} \over {\sqrt {5}}}+{\frac {1}{2}}}\right\rfloor }
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Binet%25E2%2580%2593Cauchy_identity/njSU62FW">Binet–Cauchy identity</a></li>
<li><a href="/facts/Binet_equation/Y3jkM6rb">Binet equation</a></li>
<li><a href="/facts/Cauchy%25E2%2580%2593Binet_formula/jbRCH0tj">Cauchy–Binet formula</a></li>
<li><a href="/facts/Matrix_multiplication/lYDB6Ro2">Matrix multiplication</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>John J O'Connor and Edmund F Robertson, 2005. <a href="https://web.archive.org/web/20060306000208/http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Binet.html">Jacques Philippe Marie Binet</a>. Retrieved November 21, 2005.</li>
<li>William A. McWorter Jr., 2005. <a href="http://www.cut-the-knot.org/arithmetic/Fibonacci.shtml">When the Counting Gets Tough, the Tough Count on Mathematics</a>. Retrieved November 21, 2005.</li></ul>

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

<ol>
<li id="fn:1"><p>"Jacques Philippe Marie Binet". New Catholic Dictionary. Archived from the original on 23 July 2008. Retrieved 8 June 2013. <a href="https://web.archive.org/web/20080723150838/http://saints.sqpn.com/ncd01283.htm" target="_blank">https://web.archive.org/web/20080723150838/http://saints.sqpn.com/ncd01283.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Weisstein, Eric W. "Binet's Fibonacci Number Formula". From MathWorld—A Wolfram Web Resource. Retrieved 10 January 2011. <a href="http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html" target="_blank">http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Jacques Philippe Marie Binet open-in-new

Jacques Philippe Marie Binet