Visualizing Addition on the Complex Plane

There is some mathematics that cannot be visualized in easy ways, but there are some parts of the mathematics where visualization is easy and helps with concepts. Many proofs are tedious algebraically but intuitive visually.

Here we are going to see who two complex numbers can be added and what their result looks like graphically to set the stage for further visualization of different proofs and other operations like multiplication and division.

In this interactive graph, click and drag the two blue points to change their value and position. These blue points represent the two complex numbers $z_1$ and $z_2$.

The red point represents their sum $z_1 + z_2$

We can create a vector for each point by drawing a line from the origin to each point. We see that the addition of complex numbers is simply the addition of their respective vectors using the head to tail rule.

Head to tail rule is simply the following. To add two vectors, place the second vector with its original magnitude and direction at the head (or top) of the first vector, placing the head of the first at the tail of the second. You continue for all vectors you want to add and then draw the resulting vector from the tail of the first vector to the head of the last vector. The head of this resultant vector will be the sum of vectors (or the complex numbers in this case).

This next example involves four points being added together to further cement the head to tail rule used to add complex numbers together.