Video Transcript
What complex number lies at the midpoint of the complex numbers π equals five plus three π and π equals three plus seven π when theyβre represented on a complex plane?
We know that the complex plane looks a lot like the Cartesian plane. Except this time, the horizontal axis represents the real component of the complex number, whereas the vertical axis represents the imaginary component. So this means the complex number π, which is five plus three π would have coordinates five, three, whereas the complex number π equals three plus seven π can be plotted with coordinates three, seven. We then could use the fact that the midpoint of two Cartesian coordinates, π₯ one, π¦ one and π₯ two, π¦ two, is π₯ one plus π₯ two all over two, π¦ one plus π¦ two all over two. And so the midpoint of the Cartesian coordinates 3, 7 and 5, 3 would be three plus five over two, seven plus three over two.
Well, three plus five is eight and eight over two is four, whilst seven plus three is 10 and 10 over two is five. And so in coordinate form, we see that the midpoint is four, five. But of course, we refer back to the complex plane. And we see that the real component of this complex number must, therefore, be four and its imaginary component must be five. And this means the complex number that lies at the midpoint of the complex numbers π and π is four plus five π. Now, in fact, itβs hugely useful to recall how we plot complex numbers to answer this question. But we can see that, to find the midpoint of two complex numbers, all we do is we find the average of the real components and the average of the imaginary components.
