The contribution of Argand to mathematics is an excellent example of being at the right place at the right time. Argand was not a trained mathematician, but a Parisian bookkeeper who never associated with or belonged to any group of mathematicians, amateur or otherwise. He studied mathematics as a hobby and was entirely self-taught. In 1806 he hit upon an idea that other mathematicians were working on at the same time.

Argand devised a method of graphing complex numbers using the techniques of analytic geometry. Similar techniques were being developed at the same time by Carl Friedrich Gauss and Caspar Wessell. The Norwegian-born Wessell had published his method in 1799, but the result went unnoticed, and Gauss did not publish his views until thirty years later. Meanwhile Argand first published his work privately in 1806, and later republished his method in the French journal Annales de mathematiques in 1813. Argand 's paper contains a few "firsts," namely, the first use of the word absolute to indicate the distance between two points on a graph, even if they represent negative numbers. Argand used a bar over a pair of letters for the first time to indicate what is today known as a vector. Argand also introduced the term modulus for the length of a vector that represents a complex number, thus anticipating the work of Augustin-Louis Cauchy.

Argand's work was conducted independently of other mathematicians. It was only after his paper on complex numbers was published that he began to make serious connections with other scientists. Argand continued to pursue mathematical work and publish occasional comments on articles written by others. But after making his great discovery, Argand contributed little to the field of mathematics and eventually died in obscurity in Paris in 1822.