Among the few other facts known of his life is a little information about his children. His son was born in Paris and continued to live there, while his daughter, Jeanne-Françoise-Dorothée- Marie-Elizabeth Argand, married Félix Bousquet and they lived in Stuttgart.
Argand is famed for his geometrical interpretation of the complex numbers where i is interpreted as a rotation through 90°. The concept of the modulus of a complex number is also due to Argand but Cauchy, who used the term later, is usually credited as the originator this concept. The Argand diagram is taught to most school children who are studying mathematics and Argand's name will live on in the history of mathematics through this important concept. However, the fact that his name is associated with this geometrical interpretation of complex numbers is only as a result of a rather strange sequence of events.
The first to publish this geometrical interpretation of complex numbers was Caspar Wessel. The idea appears in Wessel's work in 1787 but it was not published until Wessel submitted a paper to a meeting of the Royal Danish Academy of Sciences on 10 March 1797. The paper was published in 1799 but not noticed by the mathematical community. Wessel's paper was rediscovered in 1895 when Juel draw attention to it and, in the same year, Sophus Lie republished Wessel's paper.
This is not as surprising as it might seem at first glance since Wessel was a surveyor. However, Argand was not a professional mathematician either, so when he published his geometrical interpretation of complex numbers in 1806 it was in a book which he published privately at his own expense. His knowledge of the book trade allowed him to put out this small edition but one would have expected it to be in a less noticable place than Wessel's work which after all was published by the Royal Danish Academy. Perhaps even more surprisingly, Argand's name did not even appear on the book so it was impossible to identify the author.
The way that Argand's work became known is rather complicated. Legendre was sent a copy of the work and he sent it to François Français although neither knew the identity of the author. After François Français's death in 1810 his brother Jacques Français worked on his papers and he discovered Argand's little book among them. In September 1813 Jacques Français published a work in which he gave a geometric representation of complex numbers, with interesting applications, based on Argand's ideas. Jacques Français might easily have claimed these ideas for himself, but he did quite the reverse. He ended his paper by saying that the idea was based on the work of an unknown mathematician and he asked that the mathematician should make himself known so that he might receive the credit for his ideas.
The article by Jacques Français appeared in Gergonne's journal Annales de mathématiques and Argand responded to Jacques Français's request by acknowledging that he was the author and submitting a slightly modified version of his original work with some new applications to the Annales de mathématiques. There is nothing like an argument to bring something to the attention of the world and this is exactly what happened next. A vigorous discussion between Jacques Français, Argand and Servois took place in the pages of Gergonne's Journal. In this correspondence Jacques Français and Argand argued in favour of the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra.
One might have expected that Argand would have made no other contributions to mathematics. However this is not so and, although he will always be remembered for the Argand diagram, his best work is on the fundamental theorem of algebra and for this he has received little credit. He gave a beautiful proof (with small gaps) of the fundamental theorem of algebra in his work of 1806, and again when he published his results in Gergonne's Journal in 1813. Certainly Argand was the first to state the theorem in the case where the coefficients were complex numbers. Petrova, in , discusses the early proofs of the fundamental theorem and remarks that Argand gave an almost modern form of the proof which was forgotten after its second publication in 1813.
After 1813 Argand did achieve a higher profile in the mathematical world. He published eight further articles, all in Gergonne's Journal, between 1813 and 1816. Most of these are based on either his original book, or they comment on papers published by other mathematicians. His final publication was on combinations where he used the notation (m, n) for the combinations of n objects selected from m objects.
In  Jones sums up Argand's work as follows:-
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 recognised by some of the geniuses of his time, 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 laboured.
Article by: J J O'Connor and E F Robertson