Addition of Complex Numbers

In light of the remarks made above, let us give a geometrical significance to the operation of adding complex numbers.

Consider the following two complex numbers:

\[\begin{align}&{z_1} = {a_1} + i{b_1}\\&{z_2} = {a_2} + i{b_2}\end{align}\]

These are plotted in the complex plane below:

How do we geometrically plot the complex number \({z_1} + {z_2}\)? The first step is to think of \({z_1}\) and \({z_2}\) and vectors. If you join the origin to \({z_1}\), you have the vector \({z_1}\). Similarly, if you join the origin to \({z_2}\), you have the vector \({z_2}\):

Now, these two vectors can be added using the parallelogram law of vector addition:

The resultant vector represents the vector \({z_1} + {z_2}\). Its tip represents the point \({z_1} + {z_2}\).

Let us apply this to a specific example. Take \({z_1} = 2 + 3i\) and \({z_2} =  - 2 + i\). These two points are plotted below:

To add the two complex numbers, let’s first think of them as vectors:

Next, we add these two vectors using the parallelogram law. The resultant vector represents the vector ; its tip represents the point \({z_1} + {z_2}\), or the point \(4i\):

The addition result can be verified algebraically:

\[{z_1} + {z_2} = \left( {2 + 3i} \right) + \left( { - 2 + i} \right) = 4i\]

Example 1: Geometrically add the complex numbers \({z_1} =  - 1 + 3i\) and \({z_2} = 4 - i\).

Solution: Consider the following figure:

Clearly,

 \[{z_1} + {z_2} = 3 + 2i\]

You can verify this algebraically.