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How to Find the Midpoint of Two Complex Numbers

When you are asked to find the midpoint of a line segment $l$ in the complex plane, you can calculate the mean of the line segments’ endpoints. The mean $m$ of two complex numbers $z$ and $w$ is calculated in the same manner as for real numbers: $m = \frac{z + w}{2}$ .

A line segment between z and w in the complex plane.

Example 1

Find the midpoint of the line segment $l$

From the graph you find that the endpoints of the line segment are $z = - 2 - i$ and $w = 4 + 3 i$ . By using the endpoints you can calculate the midpoint:

\begin{array}{l} m & = \frac{z + w}{2} \\ = \frac{(- 2 - i) + (4 + 3 i)}{2} \\ = \frac{(- 2 + 4) + (- 1 + 3) i}{2} \\ = \frac{2 + 2 i}{2} \\ = 1 + i . \end{array}

A line segment and its midpoint drawn in the complex plane.

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