1. Overview
Jean-Robert Argand (Jean-Robert Argand[ʒɑ ʁɔbɛʁ aʁɡɑ]French; July 18, 1768 - August 13, 1822) was a Genevan amateur mathematician. While managing a bookstore in Paris in 1806, he privately published his groundbreaking work on the geometric interpretation of complex numbers, which introduced the concept now known as the Argand diagram. He is also recognized for presenting one of the first rigorous and complete proofs of the Fundamental Theorem of Algebra.
2. Life
Jean-Robert Argand's life was marked by self-study and significant contributions to mathematics despite a lack of formal academic affiliation.
2.1. Early Life and Background
Jean-Robert Argand was born on July 18, 1768, in Geneva, which was then the Republic of Geneva. His parents were Jacques Argand and Eve Carnac. Details regarding his early life and formal education are largely unknown. As he did not belong to any established mathematical organizations and his knowledge of mathematics was self-taught, it is believed that he pursued mathematics primarily as a hobby rather than a professional career.
2.2. Career in Paris
In 1806, Argand relocated to Paris with his family. While managing a bookstore in the city, he privately published his seminal mathematical work, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Essay on a method of representing imaginary quantities in geometric constructions). This essay, initially self-published, was later republished in 1813 in the prominent French journal Annales de Mathématiques, bringing his ideas to a wider audience.
2.3. Death
Jean-Robert Argand died in Paris on August 13, 1822. The specific cause of his death remains unknown.
3. Mathematical Contributions
Argand's principal achievements in mathematics centered on his innovative approach to complex numbers and his rigorous proof of a fundamental algebraic theorem.
3.1. Geometric Interpretation of Complex Numbers
Argand's 1806 publication, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques, introduced a revolutionary method for visualizing complex numbers using analytical geometry. This work proposed the interpretation of the imaginary unit i as a 90-degree rotation in what became known as the Argand plane or complex plane. This geometric representation provided a powerful tool for understanding complex number operations. At the time, other mathematicians, including Carl Friedrich Gauss and Caspar Wessel, were also exploring the topic of complex numbers and similar graphing techniques. Notably, Wessel's 1799 paper on a comparable graphing method did not gain significant attention upon its publication.
3.2. Magnitude and Vector Notation
In his Essai, Argand was also the first to introduce the concept of the 'modulus' to represent the magnitude of both vectors and complex numbers. This idea provided a way to measure the "length" or "size" of these mathematical entities. Furthermore, he developed a specific notation for vectors, representing them as directed line segments, which contributed to the evolving language of vector mathematics.
3.3. Proof of the Fundamental Theorem of Algebra
In 1814, Argand published another significant work titled Réflexions sur la nouvelle théorie d'analyse (Reflections on the new theory of mathematical analysis). In this publication, he presented one of the first rigorous and complete proofs of the Fundamental Theorem of Algebra. His proof was notable for being the first to generalize the theorem to include polynomials with complex coefficients, expanding its applicability beyond real coefficients.
4. Reception and Legacy
Argand's work, though initially circulated privately, eventually gained recognition and significantly influenced the development of mathematics, particularly in the understanding of complex numbers and algebraic theory.
4.1. Publication and Early Reception
Argand's initial essay on the geometric interpretation of complex numbers was privately published in 1806. Its subsequent republication in the Annales de Mathématiques in 1813 helped disseminate his ideas within the mathematical community. His proof of the Fundamental Theorem of Algebra, published in 1814, was later incorporated into influential mathematical texts. For instance, Augustin-Louis Cauchy's 1821 textbook, Cours d'analyse de l'École Royale Polytechnique, included Argand's proof, although Argand was not directly credited for it in that work. The proof was also later referenced in Chrystal's influential textbook Algebra.
4.2. Influence on Later Mathematics
Argand's innovative ideas, particularly the Argand diagram and his rigorous proof of the Fundamental Theorem of Algebra, were adopted and referenced by subsequent mathematicians. Their inclusion in foundational textbooks by figures like Cauchy and Chrystal ensured their lasting impact and integration into mainstream mathematical education and research. The geometric interpretation of complex numbers became a standard tool, simplifying complex number operations and making them more intuitive.
4.3. Modern Assessment
In 1978, Argand's proof of the Fundamental Theorem of Algebra was highlighted by The Mathematical Intelligencer, which described it as "both ingenious and profound." This modern assessment underscores the historical significance and enduring value of his contributions to the understanding of complex numbers and algebraic theory, solidifying his place as an important figure in the history of mathematics despite his amateur status.