Visualizing Multiplication of Complex Numbers on the Complex Plane

Multplication of complex numbers seems abstract. Multiplication of real numbers is well understood as the same as adding the number to itself "n" number of times. $3 \times 4$ is simply 3 being added to itself 4 times. Even $3 \times 4.5$ is understood at 3 addeded to itself 4 times and then half of 3. Similarly $3 \times \sqrt{5}$ can be understood as 3 added to itself by 2 and then some change.

But multiplication of complex numbers is too abstract at this stage. What does it mean to add a complex number $3 + 4i$. Or even, what does it mean for a number $z$ to multiply it by just $i$?

There is no easy algebraic understanding of this multiplication. But graphically, we can see this as a transformation. Consider a number $z_1$. It has a unique representation in the complex plane. How does this representation change when that number is multiplied by another complex number $z_2$?

Let us visualize this transformation in the given chart. The green point is the $z_1$ and the blue point is the $z_2$ that you can change around. The red point is the $z_1 \times z_2$. Play around changing the two points to try and understand the transformation and draw some conclusions.

Insights

There are two things to note about the transformation. The magnitude and the angle of the result.

We note that the angle of $z_1 \times z_2$ is the sum of the angles of $z_1$ and $z_2$ as measured from the x-axis. This can be written as;

Then we note that the magnitude of $z_1 \times z_2$ is also multiplicative of the two. Which we can write as;

Applications of Multiplication

Hopefully this allows you to create a more visual understanding of what it means to multiply complex numbers. It transforms the result into adding their angles and multiplying thier moduli together. In one of the advance applications, complex numbers will represent signals. So, if you would want to reduce the strength of a signal, you will have to multiply it with a complex number with modulus less than 1. Similarly, if you want to change the phase of the signal which involves pulling its angle forward or backward, you would know to multiply it with a complex number with a negative angle to pull it back and a positive angle to pull it forward.

This is oversimplified version, but you should get the gist. Multiplication transforms the numbers and it helps to have a visual and intuitive understanding of what it means to multiply two numbers.

Insights

There are again two main insights to note here.

First, the multiplicative inverse always has the negative angle, it is flipped on the x-axis such that the angles of $z$ and $z^{-1}$ cancel each other out. The two angles will be coterminal.

Second, the moduli of $z$ and $z^{-1}$ are in an inverse proportion. We note that;

This relation must be maintained. So increasing the modulus of one decreases the modulus of the other and vice versa. This also has the nice corrollary that if one complex number has the modulus of 1, the other two must also have a modulus of one. You can check it out yourself by playing with the graph and trying to make the number and the inverse equal.