Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

whose work emphasized

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

and

thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

Biography

Joseph Bertrand was the son of physician Alexandre Jacques François Bertrand and the brother of archaeologist Alexandre Bertrand. His father died when Joseph was only nine years old; by that time he had learned a substantial amount of mathematics and could speak Latin fluently. At eleven years old he attended the course of the

École Polytechnique (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...

as an auditor. From age eleven to seventeen, he obtained two bachelor's degrees, a license and a PhD with a thesis concerning the mathematical theory of electricity, and was admitted to the 1839 entrance examination of the École Polytechnique. Bertrand was a professor at the

and Collège de France, and was a member of the Paris Academy of Sciences of which he was the permanent secretary for twenty-six years. He conjectured, in 1845, that there is at least one prime number between ''n'' and 2''n'' − 2 for every ''n'' > 3. Chebyshev proved this conjecture, now termed

Bertrand's postulate In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every

n > 1

, there is always at least one ...

, in 1850. He was also famous for two paradoxes of

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

, known now as Bertrand's Paradox and the Paradox of Bertrand's box. There is another paradox concerning

game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

that is named for him, known as the Bertrand Paradox. In 1849, he was the first to define real numbers using what is now termed a

Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...

. Bertrand translated into French

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

's work concerning the theory of errors and the method of least squares. Concerning

, he reviewed the work on

oligopoly An oligopoly () is a market in which pricing control lies in the hands of a few sellers. As a result of their significant market power, firms in oligopolistic markets can influence prices through manipulating the supply function. Firms in ...

theory, specifically the Cournot Competition Model (1838) of French mathematician

Antoine Augustin Cournot Antoine Augustin Cournot (; 28 August 180131 March 1877) was a French philosopher and mathematician who contributed to the development of economics. Biography Antoine Augustin Cournot was born on August 28, 1801 in Gray, Haute-Saône. He ent ...

. His Bertrand Competition Model (1883) argued that Cournot had reached a very misleading conclusion, and he reworked it using prices rather than quantities as the strategic variables, thus showing that the equilibrium price was simply the competitive price. His boo
Thermodynamique
states in Chapter XII, that thermodynamic entropy and temperature are only defined for reversible processes. He was one of the first people to state this publicly. In 1858 he was elected a foreign member of the

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...

Works by Bertrand

*
Traité de calcul différentiel et de calcul intégral
' (Paris : Gauthier-Villars, 1864–1870) (2 volumes treatise on calculus) *
Rapport sur les progrès les plus récents de l'analyse mathématique
' (Paris: Imprimerie Impériale, 1867) (report on recent progress in mathematical analysis) *
Traité d'arithmétique
' (L. Hachette, 1849) (arithmetics) *
Thermodynamique
' (Paris : Gauthier-Villars, 1887) *
Méthode des moindres carrés
' (Mallet-Bachelier, 1855) (translation of

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

's work on least squares) *
Leçons sur la théorie mathématique de l'électricité / professées au Collège de France
' (Paris : Gauthier-Villars et fils, 1890) *
Calcul des probabilités
' (Paris : Gauthier-Villars et fils, 1889) *
Arago et sa vie scientifique
' (Paris : J. Hetzel, 1865) (biography of Arago) *
Blaise Pascal
' (Paris : C. Lévy, 1891) (biography) *
Les fondateurs de l'astronomie moderne: Copernic, Tycho Brahé, Képler, Galilée, Newton
' (Paris: J. Hetzel, 1865) (biographies)

References

External links

Bertrand, Joseph Louis Francois (1822–1900)
(archived 4 April 2004)

(archived 3 March 2016)

* *
Author profile
in the database zbMATH {{DEFAULTSORT:Bertrand, Joseph Louis Francois 1822 births 1900 deaths Scientists from Paris 19th-century French mathematicians 19th-century French economists Differential geometers French number theorists French probability theorists Members of the Académie Française Members of the French Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences École Polytechnique alumni Academic staff of the Collège de France Foreign members of the Royal Society Members of the Royal Swedish Academy of Sciences Honorary members of the Saint Petersburg Academy of Sciences Lycée Saint-Louis alumni

Biography

Works by Bertrand

See also

Further reading

References

External links