# 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 $\R^2$:

This representation is known as an **Argand diagram**.

## Also known as

Some sources refer to this as an **Argand plane**.

It is also sometimes known as a **Gauss Plane**, or **Gaussian 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.

## Source of Name

This entry was named for Jean-Robert Argand.

## Historical Note

The Argand diagram appears in Jean-Robert Argand's 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 by geometric constructions*).

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 pures et appliquées*, 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 pures et appliquées*.

By this time, however, Carl Friedrich Gauss had already himself invented the same concept.

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.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $6$: Graph of a Complex Number - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Graphical Representation of Complex Numbers - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Argand diagram (complex plane)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Argand diagram (complex plane)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Argand diagram**