# Complex Plane

The complex plane (also called the Argand plane or Gauss plane) is a way to represent complex numbers geometrically. It is basically a modified Cartesian plane, with the real part of a complex number represented by a displacement along the $x$-axis, and the imaginary part by a displacement along the $y$-axis.

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## Motivation

The motivation behind the complex plane stems from the fact that a complex number, in its essence, is just an ordered pair of real numbers. So any complex number can be given a concrete geometric interpretation as points on a plane.

The complex number $a+bi$ can simply be represented as the point on the Cartesian plane with the coordinates $(a, b)$.

This gives us a way to deal with complex numbers visually, which has a lot of advantages. Adding or multiplying complex numbers can be viewed as geometric operations on points (or vectors) on the plane if we represent complex numbers this way.

## What It Looks Like

As we said earlier, the complex plane is basically a modified Cartesian plane where the $x$-axis and the $y$-axis have been dubbed the "real axis" and the "imaginary axis," respectively.

Here we can see several complex numbers plotted on the complex plane.

## History

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), who was an amateur mathematician and a keeper of a bookstore in Paris, although they were first described by Danish land surveyor and mathematician Caspar Wessel (1745–1818).