Definition:Complex Number/Complex Plane

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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 the complex plane.


Real Axis

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

Hence the $x$-axis of the complex plane is known as the real axis.


Imaginary Axis

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

Hence the $y$-axis of the complex plane 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

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