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.
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).
The topic of complex numbers was also being studied by other mathematicians, notably Carl Friedrich Gauss and Caspar Wessel.
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).
The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821).