The Argand Diagram is a great tool to help us understand complex numbers and how they change. Let’s look at how we can use it in an easy way!

What is it?

In the Argand Diagram, we show complex numbers as points.

For example, if we have the complex number $z = 3 + 4i$, we can find it at the point $(3, 4)$ on the diagram. This point clearly shows us the real part (3) and the imaginary part (4) of the complex number.

How do they change?

Now, let’s talk about how these complex numbers can change. Here are three important ways:

1. Translation: This means moving a complex number by adding another complex number.

   • For instance, if we add $1 + 2i$ to $z$, it will move from $(3, 4)$ to $(4, 6)$.

2. Rotation: This is when we change the position of a complex number by multiplying it by a special unit complex number.

   • For example, multiplying by $e^{i\theta}$ will rotate the point around the starting point (the origin) by an angle $\theta$.

3. Scaling: This means changing how far a complex number is from the origin by multiplying it by a regular number.

   • For example, if we multiply $z$ by 2, it will move from $(3, 4)$ to $(6, 8)$.

By looking at these changes on the Argand Diagram, students can see how complex numbers work together. This helps make sense of both algebra and geometry!