# Definition:Complex Number/Complex Plane

## 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 the **complex plane**.

### Real Axis

Complex numbers of the form $\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis.

### Imaginary Axis

Complex numbers of the form $\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis.

This line is known as the **imaginary axis**.

## Also known as

Some sources refer to the **complex plane** as an **Argand plane**, for **Jean-Robert Argand**.

It is also sometimes known as a **Gauss plane**, or **Gaussian plane**, for **Carl Friedrich Gauss**.

As it is now recognised that neither Gauss nor Argand had precedence over the concept of plotting complex numbers on the cartesian plane, the more neutral term **complex plane** is usually preferred nowadays.

## Also see

- Results about
**the complex plane**can be found**here**.

## Historical Note

It is reported by Ian Stewart and David Tall, in their *Complex Analysis (The Hitchhiker's Guide to the Plane)* of $1983$, that John Wallis represented a complex number using this technique in his *A Treatise on Algebra*, but for some reason was ignored.

This has not been corroborated by $\mathsf{Pr} \infty \mathsf{fWiki}$, and there may be some doubt as to its truth, considering the given publication date of *A Treatise on Algebra* ($1673$) does not match that given by all other sources found ($1685$).

It is widely reported that the concept of the **complex plane** was an invention of Caspar Wessel, independently of Jean-Robert Argand and Carl Friedrich Gauss.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{Q}$ - 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 - 1990: H.A. Priestley:
*Introduction to Complex Analysis*(revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**complex plane** - 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur:
*Quantum Mechanics*... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**complex plane** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**complex plane**