Definition:Argand Diagram

Definition
Because a complex number can be expressed as an ordered pair, we can plot the number $x + i y$ on the real number plane $\R^2$:


 * ComplexPlane.png

This representation is known as an Argand Diagram.

It is also sometimes known as a Gauss Plane, but as it is now recognised that neither Gauss nor Argand had precedence over the concept of plotting complex numbers on a plane, the more neutral term complex plane is usually preferred nowadays.

The concept appears in his self-published 1806 work Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Essay on a method of representing imaginary quantities). This would have passed unnoticed by the mathematical community except that Legendre received a copy. He had no idea who had published it (as Argand had failed to include his name anywhere in it). Legendre passed it to François Français. His brother Jacques Français found it in his papers after his death in 1810, and published it in 1813 in the journal Annales de mathématiques, announcing it as by an unknown mathematician. He appealed for the author of the work to make himself known, which Argand did, submitting a slightly modified version for publication, again in Annales de mathématiques.

It must be noted that this concept had in fact been invented by Caspar Wessel as early as 1787, and been published in the paper Om directionens analytiske betegning by the Danish academy in 1799. This paper was rediscovered in 1895 by Sophus Juel, and later that year Sophus Lie republished it. Wessel's precedence is now universally recognised, but the term Argand Diagram has stuck.