# Complex plane

In mathematics, the **complex plane** is a way of visualising the space of the complex numbers. It can be thought of as a modified cartesian plane, with the real part represented in the x-axis and the imaginary part represented in the y-axis. The x-axis is also called the **real axis** and the y-axis is called the **imaginary axis**.

The complex plane is sometimes called the **Argand plane** for its use in **Argand diagrams**. Its creation is generally credited to Jean-Robert Argand, although it was first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, and the multiplication of complex numbers can be expressed simply using polar coordinates, where the magnitude of the product is the product of those of the terms, and the angle from the real axis of the product is the sum of those of the terms.

Argand diagrams are frequently used to plot the positions of poles and zeros of a function in the complex plane.

## Use of the complex plane in control theory

In control theory, one use of a complex plane is that known as the 's-plane'. It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the Laplace operator 's', hence the name 's' plane.

In addition, a complex plane (not the "s" plane) is used with the Nyquist stability criterion. This is a geometric principle which allows the stability of a control system to be determined from inspection of a Nyquist plot of its frequency-phase response in the complex plane.

The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation.