Find the midpoint of three plus five 𝑖 and seven minus 13𝑖.
We recall that three plus five 𝑖 and seven minus 13𝑖 are complex numbers. It’s actually really straightforward to find the midpoint of two complex numbers. But to see what’s going on, let’s 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 𝑖 will lie in the first quadrant of the complex plane, whereas the complex number seven minus 13𝑖 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 𝑥- and 𝑦-coordinates, we could extend this information to finding the midpoint of two complex numbers. We see that the midpoint of two complex numbers 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 is 𝑎 plus 𝑏 over two plus 𝑏 plus 𝑑 over two 𝑖. 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 𝑖 and seven minus 13𝑖 is three plus seven over two plus five plus negative 13 over two 𝑖. That’s 10 over two plus negative eight over two 𝑖, which is equal to five minus four 𝑖. The midpoint of three plus five 𝑖 and seven minus 13𝑖 is five minus four 𝑖.