Visualizing Complex Numbers

In the last section, we introduced complex numbers and how to perform basic operations on them.

Question: How would you graphically represent a complex number?

If you remember from a previous lesson on the Number Line and the Cartesian Plane, a single number is represented on a number line, and a two numbers together (called coordinates) are represented on the Cartesian Plane.

Generally, remember that we need as many dimensions as the components of the thing we want to represent. Take a function graph like $y = x^2$. All algenraic functions have one input and one output and so they are graphically represented in two dimesions using a cartesian plane.

Well, complex numbers also have two parts. If we want to visually or graphically represent them, we also need two dimensions, one for the real part and one dimension of the imaginary part.

Introducing the Argand Diagram

The 2D-plane used to visualize complex numbers is called the Argand Diagram. It looks exactly like a Cartesian Plane, but we give the axes new meaning:

• The horizontal axis (x-axis) represents the real part of the complex number
• The vertical axis (y-axis) represents the imaginary part

So the complex number $z_1 = 3 + 2i$ would be plotted at the point $(3, 2)$, and $z_2 = -1 - 4i$ would be at $(-1, -4)$.

Ordered Pairs and Coordinates

When we write a point as $(x, y)$, we're using what's called an ordered pair. Each pair of numbers gives us a precise location on a 2D plane:

• The first value $x$ is called the abscissa — it shows how far left or right to go (along the real axis)
• The second value $y$ is the ordinate — it tells us how far up or down to go (along the imaginary axis)

Two Ways to Think About It

There are two powerful ways to interpret this plane:

1. As a completely new kind of plane — the complex plane or Argand Diagram — made specifically to visualize complex numbers.
2. Or, as an extension of the real number line — where we've added a whole new dimension: the imaginary direction. Because the real number 2 can be attached to an infinite number of infinite numbers to make infinite complex numbers. All those complex numbers are then vertically up or down the 2 on the x-axis.

This new perspective allows us to manipulate and visualize complex numbers much like vectors — with magnitude and direction — opening up new mathematical possibilities.

In future lessons, we'll see how complex numbers can be added, multiplied, and even rotated using this visual tool — but for now, it's enough to understand that each complex number corresponds to a unique point on the Argand Diagram.