Jean-Robert Argand was a French amateur mathematician. He is noted for his original contribution to mathematics by publishing geometrical interpretation of complex numbers known as the Argand diagram.

Background

Jean-Robert Argand was born on July 18, 1768 in Geneva. Biographical data on Argand is limited but it is known that he was the son of Jacques Argand and Èves Canac, and that he was baptized on 22 July (a date given by some for his birth).

Education

Argand’s training and background are so little known that he has often been confused with a man to whom he probably was not even related, Aimé Argand, a physicist and chemist who invented the Argand lamp.

Career

Argand was a man with an unknown background, a nonmathematical occupation, and an uncertain contact with the literature of his time who intuitively developed a critical idea for which the time was right. He exploited it himself. The quality and significance of his work were recognized by some of the geniuses of his lime, but breakdowns in communication and the approximate simultaneity of similar developments by other workers force a historian to deny him full credit for the fruits of the concept on which he labored.

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. 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.

In 1978 it was called by The Mathematical Intelligencer “both ingenious and profound,” and was later referenced in Cauchy's Cours d’Analyse and Chrystal's influential textbook Algebra.

Jean-Robert Argand died of an unknown cause on August 13, 1822 in Paris.

Argand, a Parisian bookkeeper, apparently never belonged to any group of mathematical amateurs or dilettantes. He began his book, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques, with a brief discussion of models for generating negative numbers by repeated subtraction; one used weights removed from a pan of a beam balance, the other subtracted francs from a sum of money. From these examples he concluded that distance may be considered apart from direction, and that whether a negative quantity is considered real or “imaginary” depends upon the kind of quantity measured. This initial use of the word “imaginary” for a negative number is related to the mathematical-philosophical debates of the time as to whether negative numbers were numbers, or even existed. In general, Argand used “imaginary” for multiples of 1, a practice introduced by Descartes and common today. He also used the term “absolute” for distance considered apart from direction.

Argand then suggested that “setting aside the ratio of absolute magnitude we consider the different possible relations of direction” and discussed the proportions +1 : -F 1 : : — 1 : — 1 and + 1 : — 1 : : — 1 : + 1. He noted that in them the means have the same or opposite signs, depending upon whether the signs of the extremes are alike or opposite. This led him to consider 1 :x: :x: — 1. In this proportion he said that x cannot be made equal to any quantity, positive or negative; but as an analogy with his original models he suggested that quantities which were imaginary when applied to “certain magnitudes” became real when the idea of direction was added to the idea of absolute number.

Argand’s notation in his original essay is of particular interest because it anticipated the more abstract and modern ideas, later expounded by W. R. Hamilton, of complex numbers as arbitrarily constructed new entities defined as ordered pairs of real numbers. This modern aspect of Argand’s original work has not been generally recognized. Argand recognized the nonrigorous nature of his reasoning, but he defined his goals as clarifying thinking about imaginaries by setting up a new view of them and providing a new tool for research in geometry. He used complex numbers to derive several trigonometric identities, to prove Ptolemy’s theorem, and to give a proof of the fundamental theorem of algebra.

Argand’s work contrasts with Wessel’s in that the latter’s approach was more modern in its explicit use of definitions in setting up a correspondence between a + bsf^l and vectors referred to a rectangular coordinate system (which neither Wessel nor Argand ever explicitly mentioned or drew). Wessel stressed the consistency of his assumptions and derived results without regard for their intuitive validity. He did not present as many mathematical consequences as Argand did.

Just as it seems clear that Argand’s work was entirely independent of Wessel’s, so it also seems clear that it was independent of the algebraic approach published by Suremain de Missery in 1801. Argand refuted the suggestion that he knew of Buée’s work published in the Philosophical Transactions of the Royal Society in 1806 by noting that since academic journals appear after the dates which they bear, and that his book was printed in the same year the journal was dated, he could not have known of Buée’s work at the time he wrote the book. Buée’s ideas were not as clear, extensive, or well developed as Argand’s.

There are obvious connections between Argand’s geometric ideas and the later work of Moebius, Bellavitis, Hermann Grassmann, and others, but in most eases it is as difficult to establish direct out growths of his work as it is to establish that he consciously drew on Wallis, de Moivre, or Euler.

Two of the most important mathematicians of the early nineteenth century, Cauchy and Hamilton, took care to note the relationship of Argand’s work to some of their own major contributions, but claimed to have learned of his work only after doing their own. Gauss probably could have made a similar statement, but he never did. Cauchy mentioned Argand twice in his “Mémoire sur les quantités géométriques,” which appeared in Exercices d’analyse et de physique mathématique (1847).

Argand’s later publications, all of which appeared in Gergonne’s Annales, are elaborations of his book or comments on articles published by others. His first article determined equations for a curve that had previously been described in the Annales. Argand went on to suggest an application of the curve to the construction of a thermometer shaped like a watch. His analysis of probable errors in such a mechanism showed familiarity with the mechanics of Laplace, as presented in Exposition du système du monde.

His last article appeared in the volume of Annales dated 1815-1816 and dealt with a problem in combinations. In it Argand devised the notation (m,n) for the combinations of m things taken n at a time and the notation Z(m,n) for the number of such combinations.

Connections

Jean-Robert Argand had a son who lived in Paris and a daughter, Jeanne-Françoise-Dorothée-Marie-Élizabeth, who married Félix Bousquet and lived in Stuttgart.