Can You Picture Multiplying Complex Numbers on the Complex Plane?

Understanding how to multiply complex numbers on the complex plane can be tough for students.

So, what is the complex plane?

It's a flat space where we can represent complex numbers.

• The horizontal axis shows the real part of the number.
• The vertical axis shows the imaginary part.

When we multiply complex numbers, some tricky changes happen that aren’t easy to see right away.

1. What is a Complex Number?

A complex number looks like this: $z = a + bi$.

Here, $a$ is the real part and $b$ is the imaginary part.

Every complex number can be shown as a point on the complex plane, which makes it easier to picture at first.

2. How Do We Multiply Complex Numbers?

When we multiply two complex numbers, $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$, the result is:

$$z_1 \cdot z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i.$$

This means we get a new complex number.

It combines both the real and imaginary parts in a way that includes rotation and scaling:

• For the real part, we subtract the product of the imaginary parts.
• For the imaginary part, we add a mix of both the real and imaginary parts from the two numbers.

3. What About Rotation and Scaling?

When we multiply by a complex number, we can think of it in two ways:

• Rotation: The angle of the new complex number in the complex plane comes from adding the angles of the original numbers.

• Scaling: The size of the new complex number is the product of the sizes (magnitudes) of the two complex numbers.

Even though these ideas about angles and sizes can help, students still find them challenging.

Switching between rectangular coordinates (real and imaginary parts) and polar coordinates (size and angle) can make things even more confusing.

To make it easier to understand, it's really important to practice visualizing these multiplications.

Using graphing tools or software can help students see how the numbers change.

Also, doing hands-on activities and learning together can help clear things up.

Sharing experiences and using visuals can make multiplying complex numbers a lot easier to grasp!