Jacques Philippe Marie Binet

Jacques Philippe Marie Binet (French: [binɛ]; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as Binet's theorem. He is also recognized as the first to describe the rule for multiplying matrices in 1812, and Binet's formula expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier.

Jacques Philippe Marie Binet
Born	(1786-02-02)2 February 1786 Rennes, France
Died	12 May 1856(1856-05-12) (aged 70) Paris, France
Scientific career
Fields	Mathematics, physics, and astronomy

Career

Binet graduated from l'École Polytechnique 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 King Louis-Philippe of France, probably because of Binet's strong support of the previous King, Charles X. In 1823 Binet succeeded Delambre in the chair of astronomy at the Collège de France.^[1] He was made a Chevalier in the Légion d'Honneur in 1821, and was elected to the Académie des Sciences in 1843.

Binet's Fibonacci number formula

Main article: Fibonacci number § Closed-form expression

The Fibonacci sequence is defined by

• ${\displaystyle u_{0}=0}$
• ${\displaystyle u_{1}=1}$
• ${\displaystyle u_{n}=u_{n-1}+u_{n-2},{\text{ for }}n>1}$

Binet's formula provides a closed-form expression for the ${\displaystyle n^{\text{th}}}$ term in this sequence:

${\displaystyle u_{n}={\frac {(1+{\sqrt {5}})^{n}-(1-{\sqrt {5}})^{n}}{2^{n}{\sqrt {5}}}}}$      ^[2]

Given:

${\displaystyle \varphi ={(1+{\sqrt {5}}) \over 2}}$

a simplified version of Binet's formula is:

${\displaystyle u_{n}=\lfloor {{\varphi ^{n} \over {\sqrt {5}}}+{\frac {1}{2}}}\rfloor }$.

See also

• Binet–Cauchy identity
• Binet equation
• Cauchy–Binet formula
• Matrix multiplication

Notes

1. "Jacques Philippe Marie Binet". New Catholic Dictionary. Archived from the original on 23 July 2008. Retrieved 8 June 2013.
2. Weisstein, Eric W. "Binet's Fibonacci Number Formula". From MathWorld—A Wolfram Web Resource. Retrieved 10 January 2011.

References

• John J O'Connor and Edmund F Robertson, 2005. Jacques Philippe Marie Binet. Retrieved November 21, 2005.
• William A. McWorter Jr., 2005. When the Counting Gets Tough, the Tough Count on Mathematics. Retrieved November 21, 2005.