A complex number $z$ is defined as an ordered pair $(x,y)$ of real numbers. In the Cartesian plane, every ordered pair of real numbers can be represented by a point. Hence, the complex number $z=(x,y)$ can be represented with a point $P$ in the Cartesian plane with coordinates $(x,y)$. Thus, every point in the plane represents a complex number and is called the **complex plane**.

When complex numbers are represented by points in the Cartesian plane, the plane is called the complex plane; and the set of such points form an **Argand diagram**. The idea of expressing complex numbers geometrically was formulated by Argand (French) and Gauss (German). The name “*complex number*” is due to Gauss.

Let $z=(x,y)=x+iy$ be a complex number represented by a point $P$ in the complex plane. Then, the modulus of the complex number $z$ is given by \[|z|=\sqrt{x^2+y^2}=OP\]

In other words, the modulus of a complex number is the distance of the point representing the complex number from the origin in the complex plane.

## The Unit Circle

The unit circle is the set of all points in the complex numbers with absolute value $1$. We know that $1$ is the absolute value of both $1$ and $-1$. It is also the absolute value of $i$ and $-i$ since they are both one unit away from $0$ on the imaginary axis.

Hence, the unit circle is the circle of radius $1$ centered at $0$. It includes all complex numbers of absolute value $1$, so the equation of unit circle is $|z|=1$.

**Previous:** The Cube Roots of Unity