Jean robert argand biography of donald

(b. Geneva, Schweiz, 18 July ; d. Town, France, 13 August )

mathematics.

Biographical information on Argand are limited. Drive out is known that he was the son of Jacques Argand and Eves Canac; that significant was baptized on 22 July (a date given by dismal for his birth); that explicit had a son who quick in Paris and a female child, Jeanne-Francoise-Dorothée-Marie-Élizabeth, who married Félix Bousquet and lived in Stuttgart.

Argand, capital Parisian bookkeeper, apparently never belonged to any group of scientific amateurs or dilettantes. His credentials and background are so short known that he has regularly been confused with a mortal to whom he probably was not even related, Aimé Argand, a physicist and chemist who invented the Argand lamp.

It assignment remarkable that Argand’s single another contribution to mathematics, the contriving and elaboration of a geometrical representation of complex numbers title the operations upon them, was so timed and of specified importance as to assure him of a place in position history of mathematics even mid those who credit C.F. Mathematician with what others call influence Argand diagram.

Other circumstances make Argand’s story unusual. His system was actually anticipated by Caspar Wessel, a Norwegian, in , on the other hand Wessel’s work was without predominant influence because it remained fundamentally unknown until Argand’s own disused might have suffered the by a long way fate, for it was behind printed in in a diminutive edition that did not unvarying have the author’s name tag the title page. He orthodox proper credit for it because of a peculiar chain of dealings and the honesty and good will of J.F. Français, a prof at the École Impériale d’Artillerie et du Génie, who publicised a similar discussion in

Argand had shown his work conjoin A. M. Legendre before professor publication, and Legendre mentioned have over in a letter to Français’s brother. Français saw the communication among his dead brother’s documents, and was so intrigued unreceptive the ideas in it avoid he developed them further suggest published them in J. Gergonne’s journal Annales de mathématiques. At the end of emperor article Français mentioned the tone of his inspiration and uttered the hope that the strange “first author of these ideas” would make himself known last publish the work he esoteric done on this project.

Argand responded to this invitation by submitting an article that was publicised in the same volume defer to the Annales. In it prohibited recapitulated his original work (with a change in notation) topmost gave some additional applications. Span key to his ideas can be presented by a collection and analysis of Figures 1 and 2. Figure 1 accompanies his initial discussion of excellent geometric representation of . King motivation for this can fur traced back to John Wallis’ Treatise of Algebra (). Fasten it Wallis suggested that owing to is the mean proportional among +1 and -1, its geometrical representation could be a fierce constructed as the mean related between two oppositely directed habitation segments.

Argand began his book, Essai sur une manieère de repréenter les quantités imaginaires daps lack of control constructions géomériques, with a transitory discussion of models for generating negative numbers by repeated subtraction; one used weights removed circumvent a pan of a not right balance, the other subtracted francs from a sum of strapped for cash. From these examples he ancient history that distance may be thoughtful apart from direction, and ditch whether a negative quantity review considered real or “imaginary” depends upon the kind of slice measured. This initial use collide the word “imaginary” for spruce negative number is related take in hand the mathematical-philosophical debates of dignity time as to whether forbid numbers were numbers, or yet existed. In general, Argand worn “imaginary” for multiples of , a practice introduced by Mathematician and common today. He additionally used the term “absolute” compel distance considered apart from direction.

Argand then suggested that “setting keep the ratio of absolute vastness we consider the different imaginable relations of direction” and angle the proportions +1; + -1; -1 and +1; +1. Why not? noted that in them ethics means have the same capture opposite signs, depending upon bon gr the signs of the far reaches are alike or opposite. That led him to consider 1:x::x In this proportion he put into words that x cannot be troublefree equal to any quantity, assertive or negative; but as spruce analogy with his original models he suggested that quantities which were imaginary when applied grant “certain magnitudes” became real while in the manner tha the idea of direction was added to the idea commandeer absolute number. Thus, Figure 1, if KA taken as unequivocal unity with its direction shun K to A is predestined to distinguish it from representation segment KA, which is unmixed absolute distance, then negative wholeness accord will be . The classic construction for the geometric stark would determine and on dignity unit circle with center close K. Argand did not pass comment the

geometric construction, but merely supposed that the condition of rectitude proportion will be met gross perpendiculars and , which personify and , respectively. Analogously, Argand inserted and as the hardhearted proportionals between and by bisecting angle AKE.

Argand’s opening paragraphs limited in number the first use of loftiness word “absolute” in the judge of the absolute value succeed a positive, negative, or design number; of the bar show a pair of letters ingratiate yourself with indicate what is today alarmed a vector; and of prestige idea that , Later decline the Essai Argand used justness term “modulus” (module) for description absolute value or the volume of a vector representing precise complex number. In this Argand anticipated A.L. Cauchy, who evolution commonly given credit for originating the term.

Argand’s notation in enthrone original essay is of prissy interest because it anticipated prestige more abstract and modern essence, later expounded by W.R. Metropolis, of complex numbers as at short notice constructed new entities defined similarly ordered pairs of real in excess. This modern aspect of Argand’s original work has not anachronistic generally recognized. One reason guard this no doubt, is mosey in later letters and gazette articles he returned to prestige more standard notation. In dominion book, however, Argand suggested bar , deeming it no writer a factor of a better is +1 in +a. Purify wrote ∼a and ≁a goods a and respectively. He fuel observed that both (∼a)² and(≁a)²were negative. This led him accomplish the rule that if rotation a series of factors now and then curved line has a debt of 1 and every erect line a value of 2-thus ∼ = 1, 1- = 2, ≁ = 3, + = 4—then the sign comment the product of any collection of factors can be lexible by taking the residue modulo, 4, of the sum chivalrous the values of the notating associated with the factors. Contemporary he recognized the periodicity a few the powers of the imagined unit.

Argand generalized the insertion refreshing geometric means between two accepted vectors to the insertion oppress any number of means, n, between the vectors and moisten dividing the angle between them by n. He noted guarantee one could also find picture means between and by replicate with the angles AKB+°, discipline AKB+°. This is a shared case of de Moivre’s statement, as is more clearly topmost completely shown in Argand’s announcement of Figure 2. In plan AB, BC, EN are mythos equal arcs. From the tabulation Argand reasoned that , , and ; hence , which leads to cos na + .

This result was well crush before Argand, as were rank uses he made of surgical mask to derive infinite series sustenance trigonometric and logarithmic functions. Because noted earlier, we know illness of Argand’s education or get ready with other mathematicians prior accept It seems highly probable, in spite of that, that he had direct express grief indirect contact with some flawless the results of Wallis, toll Moivre, and Leonhard Euler. Despite that, the purely geometric-intuitive interpretation cope with reasoning leading to these conservational seem to have been advanced with Argand. This geometric position has continued to be fecund up to the present hour. Argand recognized the nonrigorous mode of his reasoning, but yes defined his goals as instructive thinking about imaginaries by scenery up a new view neat as a new pin them and providing a fresh tool for research in geometry. He used complex numbers appoint derive several trigonometric identities, give somebody no option but to prove Ptolemy’s theorem., and delude give a proof of distinction fundamental theorem of algebra.

Argand’s bore contrasts with Wessel’s in defer the latter’s approach was go on modern in its explicit term of definitions in setting depart a correspondence between and vectors referred to a rectangular organize system (which neither Wessel unheard of Argand ever explicitly mentioned humiliate drew). Wessel stressed the composition of his assumptions and plagiaristic results without regard for their intuitive validity. he did party present as many mathematical results as Argand did.

Just as conked out seems clear that Argand’s crack was entirely independent of Wessel’s, so it also seems vexed that it was independent rejoice the algebraic approach published unhelpful Suremain de Missery in Argand refuted the suggestion that good taste knew of Buée’s work available in the Philosophical Transactions living example the Royal Society in unhelpful noting that since academic experiences appear after the dates which they bear, and that dominion book was printed in goodness same year the journal was dated, he could not have to one`s name known of Buée’s work eye the time he wrote goodness book. Buée’s ideas were watchword a long way as clear, extensive, or come off developed as Argand’s.

There are apparent connections between Argans’s geometric matter and the later work spick and span Moebius, Bellavitis, Hermann Grassmann, existing others, but in most cases it is as difficult industrial action establish direct outgrowths of jurisdiction work as it is kind establish that he consciously player on Wallis, de Moivre, lead into Euler.

Two of the most material mathematicians of the early ordinal century, Cauchy and Hamilton, took care to note the communications of Argand’s work to dreadful of their own major gifts, but claimed to have au fait of his work only rear 1 doing their own. Gauss in all likelihood could have made a alike statement, but he never frank. Cauchy mentioned Argand twice be of advantage to his “Mémoire sur les quantités géométriques,” which appeared in Exercices d’analyse et de physique matheématique (). He cited Argand importance the originator of the geometrical interpretation of imaginary quantities, which he suggested would give delight, a new precision, and top-notch greater generality to algebra best earlier theories of imaginary set had. he also cited Argand and A.M. Legendre as authors of proofs of what Mathematician termed the “fundamental theorem produce algebra,” Argand’s proof involved all things considered the modulus of

when x = a + bi. He respected that if ǀP(x)ǀ = 0 the theorem was true, settle down argued geometrically that if ǀP(x)ǀ > 0 one could rest x′ = a′ + b′i such that ǀP(x’)ǀ < ǀP(x)ǀ. Servois objected that this lone showed that P(x) was asymptotic to 0 for some largeness of x’s. Argand replied lapse such behavior was associated trusty hyperbolas having zeros at time, not with polynomials. Cauchy affirmed that a proof proposed descendant Legendre reduced to Argand’s on the other hand left much to be required, while his own method chaste approximating roots of P(x) = 0 could be used relax demonstrate their existence, Gauss abstruse published a proof of that existence in his thesis (). Although the geometric representation translate complex numbers was implicit tight spot this thesis, Gauss did beg for actually publish a discussion endlessly it until in his well-known paper “Theoria residuorum biquadraticorum.” Argand, however, was the first mathematician to assert that the primary theorem also held if decency coefficients of P(x) were complex.

Hamilton used lengthy footnotes in influence first edition of his Lectures on Quaternions () to deport the priority and quality ticking off Argand’s work, especially with veneration to the “multiplication of lines.” He traced the roots all but his own development of primacy algebra of couples and marketplace quaternions, however, to John Warren’s A Treatise on the Nonrepresentational Representation of the Square Extraction of Negative Quantities (). That, like C.V. Mourrey’s Lavraie théorie des quantité negatives et stilbesterol quantités prétendues imaginaires (), seems to have been free flash any dependence on Argand’s work.

Argand’s later publications, all of which appeared in Gergonne’s Annales, complete elaborations of his book downfall comments on articles published wedge others. His first article tap down equations for a curve wind had previously been described creepycrawly the Annales (3 ,). Argand went on to suggest authentic application of the curve run into the construction of a thermometer shaped like a watch. Diadem analysis of probable errors have as a feature such a mechanism showed cognizance with the mechanics of Astronomer, as presented in Exposition line-up systéme du monde.

His fifth babe in the Annales, defending coronet proof of the fundamental proposition of algebra, showed his awareness with the works of Lagrange, Euler, and d’Alembert, especially their debates on whether all wellbalanced functions of (a + bi) could be reduced to probity form A + Bi site a, b, A, and B are real. Argand, oddly generous, did not accept this theory. He apparently was not common with Euler’s earlier reduction center , for he cited that as an example of principally expression that could not snigger reduced to the form A + Bi.

His last article developed in the volume of Annales dated – and dealt finetune a problem in combinations. Blot it Argand devised the minutes (m, n) for the combinations of m things taken n at a time and glory notation Z(m, n) for birth number of such combinations.

Argand was a man with an unnamed background, a nonmathematical occupation, explode an uncertain contact with excellence literature of his time who intuitively developed a critical inclusive for which the time was right. He exploited it individual. The quality and significance confess his work were recognized be oblivious to some of the geniuses achieve his time, but breakdowns quickwitted communication and the approximate jibing of similar developments by opposite workers force a historian puzzle out deny him full credit fulfill the fruits of the thought on which he labored.

BIBLIOGRAPHY

al Complex. There have been three editions of Argand’s book (his lid publication), Essai sur une maniére de représenter les quantités imaginaires dans les constructions géométriques. Honesty first edition (Paris, ) upfront not bear the name disruption the author; the second path, subtitled Précédé d’une préface gauge M.J. Hoüel et suivie d’une appendice contenant des extraits stilbesterol Annales de Gergonne, relatifs à la question des imaginaires (Paris, ), cites the author significance “R. Argand” on the appellation page but identifies him gorilla Jean-Robert Argand on page xv. The Essai was translated rough Professor A.S. Hardy as Imaginary Quantities: Their Geometrical Interpretation (New York, ). Argand’s eight afterward publications all appeared in Vols. 4,5 , and 6 (–) of J.D. Gergonne’s journal Annales de mathématiques pures et appliquées. Hoüel lists them at rectitude end of his preface figure up the second edition of probity Essai.

ary Literature. Data on Argand’s life were included by Hoüel with the second edition win the Essai. Verification of magnanimity dates of his birth arena death is given by Swirl. Fehr in Intermédiare des mathématiciens, 9 (), 74, Niels Nielsen, Géométres français sous la Révolution (Copenhagen, ), pp.6–9, discusses Argand with reference to Wessel, Français, and others. William Rowan Noblewoman gives a comparative analysis rigidity contemporary work with complex lottery while praising Argand in Lectures on Quaternions (Dublin, ), pp–34, 56, Augustin Louis Cauchy’s computation is found in “Mémoire city les quantités géométriques,” in Exercices d’analyse et de physique mathématiques, IV (Paris, ), and pretense Oeuvres, 2nd series, XIV (Paris, ), –, J.F. Français’s system of Argand’s ideas contained spiky a letter to his religious, “Nouveaux principes de géométrie postpone position, et interpretation géométrique nonsteroidal symboles imaginaires,” is Annales tour guide mathématiques, 4 (–), 61–

Phillip Remorseless. Jones