Jacques Philippe Marie Binet

Key takeaways

Jacques Philippe Marie Binet belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Jacques Philippe Marie Binet to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Jacques Philippe Marie Binet from memory before moving on to harder problems.

Reference excerpt

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.

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. 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

The Fibonacci sequence is defined by

u 0 = 0 {\displaystyle u_{0}=0}

u 1 = 1 {\displaystyle u_{1}=1}

u n = u n − 1 + u n − 2 , for n > 1 {\displaystyle u_{n}=u_{n-1}+u_{n-2},{\text{ for }}n>1}

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

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}}}}} Given:

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

a simplified version of Binet's formula is:

u n = ⌊ φ n 5 + 1 2 ⌋ {\displaystyle u_{n}=\left\lfloor {{\varphi ^{n} \over {\sqrt {5}}}+{\frac {1}{2}}}\right\rfloor } .

See also Binet–Cauchy identity Binet equation Cauchy–Binet formula Matrix multiplication

Notes

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.

Illustrations

Jacques Philippe Marie Binet illustration

Worked examples

Example 1 — a first encounter with Jacques Philippe Marie Binet

Start with the simplest possible case. Write down what Jacques Philippe Marie Binet claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jacques Philippe Marie Binet before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jacques Philippe Marie Binet ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jacques Philippe Marie Binet

In research: Jacques Philippe Marie Binet appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Jacques Philippe Marie Binet in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Jacques Philippe Marie Binet is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1786 births, 1856 deaths, 19th-century French mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Jacques Philippe Marie Binet outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Jacques Philippe Marie Binet in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jacques Philippe Marie Binet means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Jacques Philippe Marie Binet out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jacques Philippe Marie Binet in simple terms?

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 lat…

Why does Jacques Philippe Marie Binet matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jacques Philippe Marie Binet?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jacques Philippe Marie Binet.