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Jean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a
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in
Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Si ...
, he published the idea of geometrical interpretation of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s known as the Argand diagram and is known for the first rigorous
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a c ...
of the
Fundamental Theorem of Algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
.


Life

Jean-Robert Argand was born in
Geneva Geneva ( ; french: Genève ) frp, Genèva ; german: link=no, Genf ; it, Ginevra ; rm, Genevra is the second-most populous city in Switzerland (after Zürich) and the most populous city of Romandy, the French-speaking part of Switzerland. Situa ...
, then
Republic of Geneva The Canton of Geneva, officially the Republic and Canton of Geneva (french: link=no, République et canton de Genève; frp, Rèpublica et canton de Geneva; german: Republik und Kanton Genf; it, Repubblica e Cantone di Ginevra; rm, Republica e ...
, to Jacques Argand and Eve Carnac. His background and education are mostly unknown. Since his knowledge of mathematics was self-taught and he did not belong to any mathematical organizations, he likely pursued mathematics as a hobby rather than a profession. Argand moved to Paris in 1806 with his family and, when managing a bookshop there, privately published his ''Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques'' (Essay on a method of representing imaginary quantities). In 1813, it was republished in the French journal ''Annales de Mathématiques''. The Essay discussed a method of graphing complex numbers via analytical geometry. It proposed the interpretation of the value as a rotation of 90 degrees in the Argand plane. In this essay he was also the first to propose the idea of modulus to indicate the magnitude of vectors and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, as well as the notation for vectors \overrightarrow. The topic of complex numbers was also being studied by other mathematicians, notably
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and
Caspar Wessel Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a Danish– Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the comp ...
. Wessel's 1799 paper on a similar graphing technique did not attract attention. Argand is also renowned for delivering a proof of the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
in his 1814 work ''Réflexions sur la nouvelle théorie d'analyse'' (Reflections on the new theory of
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
). It was the first complete and rigorous proof of the theorem, and was also the first proof to generalize the fundamental theorem of algebra to include
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s with complex
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s. The first textbook containing a proof of the theorem was
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
's '' Cours d'analyse de l'École Royale Polytechnique'' (1821). It contained Argand's proof, although Argand is not credited for it. And the proof was later referenced in Chrystal's influential textbook ''Algebra''. Argand died of an unknown cause on August 13, 1822 in Paris. In 1978 his proof of the fundamental theorem of algebra was called by The Mathematical Intelligencer “both ingenious and profound.”


See also

*
Caspar Wessel Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a Danish– Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the comp ...
* ''i'', the imaginary square root of −1 *
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
*
Complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
(also known as Argand plane)


References

* Roy, J. (n.d.) James Robert Argand Biography , World of Mathematics. ''Bookrags.com.'' Retrieved March 18, 2008. From http://www.bookrags.com/biography/jean-robert-argand-wom/. *McGrath, K., Travers B., et al. (n.d.) James Robert Argand Biography , Word of Scientific Discovery. ''Bookrags.com.'' Retrieved March 18, 2008. From http://www.bookrags.com/biography/jean-robert-argand-wsd/.


External links

* * Robert Argand, ''Essai sur une manière de représenter des quantités imaginaires dans les constructions géométriques'', 2e édition, Gauthier Villars, Paris (1874
BNF

Jean-Robert Argand, Biography on ''s9.com''

Imaginary quantities; their geometrical interpretation
English translation of Jean-Robert Argand's original French work


Further reading

* {{DEFAULTSORT:Argand, Jean-Robert 1768 births 1822 deaths 18th-century scientists from the Republic of Geneva Amateur mathematicians 19th-century Swiss mathematicians Complex numbers Mathematicians from the Republic of Geneva