Argand diagram

We can plot complex numbers as points on a graph which which we will call an Argand diagram by using the $x$-axis to represent the real part of the complex number and the $y$-axis to represent the imaginary part.

A complex number $z=x+y\mathrm{i}$ can be represented on an Argand diagram by a point $A(x,y)$.

For example, the complex number $z=-2+3\mathrm{i}$ can be represented on an Argand diagram by a point $B$ with coordinates $(-2,3).$

Examples

Polar form

A complex number $z=x+y\mathrm{i}$ is said to be in cartesian form as $x$ and $y$ corresponds to the $x$ and $y$ coordinates of the point on an Argand diagram.

Instead of specifying a complex number by $x$ and $y,$ we can also specify two quantities: the modulus $|z|=r$ and argument $\arg(z)=\theta$ where $r$ can be thought of as the distance from the origin to the point representing the complex number and $\theta$ is the angle between the positive real-axis and the line from the origin to the point representing the complex number.

Visualization