### Video Transcript

Find the midpoint of three plus five π and seven minus 13π.

We recall that three plus five π and seven minus 13π are complex numbers. Itβs actually really straightforward to find the midpoint of two complex numbers. But to see whatβs going on, letβs sketch each complex number on the complex plane. We recall that the complex plane looks a lot like the Cartesian plane. Except the horizontal axis represents the real component of the complex number, whereas the vertical axis represents the imaginary component. This means the complex number three plus five π will lie in the first quadrant of the complex plane, whereas the complex number seven minus 13π will lie in the fourth quadrant as shown.

Now, if we know that the midpoint of two Cartesian coordinates is the arithmetic average of their π₯- and π¦-coordinates, we could extend this information to finding the midpoint of two complex numbers. We see that the midpoint of two complex numbers π plus ππ and π plus ππ is π plus π over two plus π plus π over two π. Essentially, we find the average of their real components and the average of their imaginary components.

So, this means the midpoint of three plus five π and seven minus 13π is three plus seven over two plus five plus negative 13 over two π. Thatβs 10 over two plus negative eight over two π, which is equal to five minus four π. The midpoint of three plus five π and seven minus 13π is five minus four π.