Subtraction of Complex Numbers

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As you may expect now, the subtraction of complex numbers is geometrically similar to vector subtraction. Once again, we take two complex numbers \({z_1}\) and \({z_2}\):

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

To geometrically evaluate \({z_1} - {z_2}\), we first plot these two numbers in the complex plane, and treat them as vectors:

Complex numbers as vectors graph 1

We need to find \({z_1} - {z_2}\), or \({z_1} + \left( { - {z_2}} \right)\). In other words, we need to find the vector sum of \({z_1}\) and \( - {z_2}\). Thus, we reverse \({z_2}\) (to obtain \( - {z_2}\)), and then add \({z_1}\) and \( - {z_2}\) using the parallelogram law of vector addition:

      Complex numbers subtraction example 1

The resultant vector represents the vector \({z_1} - {z_2}\), while its tip represents the point \({z_1} - {z_2}\).

Example 1: From \({z_1} =  - 2 + 3i\), subtract \({z_2} =  - 4 + i\). Solution. First, we plot these two points in the plane, and treat them as vectors:

 Complex numbers as vectors example 1

Then, we reverse vector \({z_2}\), and add \({z_1}\) and \( - {z_2}\) using the parallelogram law:

Complex numbers subtraction example 2

The resultant vector represents \({z_1} - {z_2}\), while its tip represents the point \({z_1} - {z_2}\), which is \(2 + 2i\).

We can verify this subtraction result algebraically:

\[\begin{align}&{z_1} - {z_2} = \left( { - 2 + 3i} \right) - \left( { - 4 + i} \right)\\&= \left( { - 2 + 4} \right) + i\left( {3 - 1} \right) = 2 + 2i\end{align}\]

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