Jean-Robert Argand (UKˈɑːrɡænd, USˌɑːrˈɡɑːn(d), ʒɑ̃ ʁɔbɛʁ aʁɡɑ̃; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is known for the first rigorous proof of the Fundamental Theorem of Algebra.
Jean-Robert Argand was born in Geneva, then Republic of Geneva, 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 i 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 numbers, as well as the notation for vectors . The topic of complex numbers was also being studied by other mathematicians, notably Carl Friedrich Gauss and Caspar Wessel. 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 in his 1814 work Réflexions sur la nouvelle théorie d'analyse (Reflections on the new theory of analysis). 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 polynomials with complex coefficients.
The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821).
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The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation . Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is . Imaginary numbers are an important mathematical concept; they extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra).
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.
In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings.