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# Distance & midpoint of complex numbers

## Video transcript

Voiceover:So we have two complex numbers here. The complex number z is equal to two plus three i and the complex number w is equal to negative five minus i. What I want to do in this video is to first plot these two complex numbers on the complex plane and then think about what the distance is between these two numbers on the plane and what complex number is exactly halfway between these two numbers or another way of thinking about it, what complex number is the midpoint between these two numbers. So I encourage you to pause this video and think about it on your own before I work through it. So let's first try to plot these on the complex planes. So let me draw, so right over here, let me draw our imaginary axis. So our imaginary axis, and over here let me draw our real axis. Real axis right over there, and let's first, let's see, we're gonna have it go as high as positive two in the real axis and as low as negative five along the real axis so let's go one, two, three, four, five. One, two, three, four, five. Along the imaginary axis we go as high as positive three and as low as negative one. So we could do one, two, three and we could do one, two, three and of course I could keep going up here just to have nice markers there although we won't use that part of the plane. Now let's plot these two points. So the real part of z is two and then we have three times i so the imaginary part is three. So we would go right over here. So this is two and this is three right over here. Two plus three i, so that right over there is z. Now let's plot w, w is negative five. One, two, three, four, five, negative five minus i, so this is negative one right over here. So minus i, that is w. So first we can think about the distance between these two complex numbers; the distance on the complex plane. So one way of thinking about it, that's really just the distance of this line right over here. And to figure that out we can really just think about the Pythagorean theorem. If you hear about the Distance Formula in two dimensions, well that's really just an application of the Pythagorean theorem, so let's think about that a little bit. So we can think about how much have we changed along the real axis which is this distance right over here. This is how much we've changed along the real axis. And if we're going from w to z, we're going from negative 5 along the real axis to two. What is two minus negative 5? Well it's seven, if we go five to get to zero along the real axis and then we go two more to get to two, so the length of this right over here is seven. And what is the length of this side right over here? Well along the imaginary axis we're going from negative one to three so the distance there is four. So now we can apply the Pythagorean theorem. This is a right triangle, so the distance is going to be equal to the distance. Let's just say that this is x right over here. x squared is going to be equal to seven squared, this is just the Pythagorean theorem, plus four squared. Plus four squared or we can say that x is equal to the square root of 49 plus 16. I'll just write it out so I don't skip any steps. 49 plus 16, now what is that going to be equal to? That is 65 so x, that's right, 59 plus another 6 is 65. x is equal to the square root of 65. Now let's see, 65 you can't factor this. There's no factors that are perfect squares here, this is just 13 times five so we can just leave it like that. x is equal to the square root of 65 so the distance in the complex plane between these two complex numbers, square root of 65 which is I guess a little bit over eight. Now what about the complex number that is exactly halfway between these two? Well to figure that out, we just have to figure out what number has a real part that is halfway between these two real parts and what number has an imaginary part that's halfway between these two imaginary parts. So if we had some, let's say that some complex number, let's just call it a, is the midpoint, it's real part is going to be the mean of these two numbers. So it's going to be two plus negative five. Two plus negative five over two, over two, and it's imaginary part is going to be the mean of these two numbers so plus, plus three minus one. Three minus one, minus one, over two times i and this is equal to, let's see, two plus negative five is negative three so this is negative 3/2 plus this is three minus 1 is negative, is negative two over two is let's see three, make sure I'm doing this right. Three, something in the mean, three minus one is two divided by two is one, so three plus three. Negative 3/2 plus i is the midpoint between those two and if we plot it we can verify that actually makes sense. So real part negative 3/2, so that's negative one, negative one and a half so it'll be right over there and then plus i so it's going to be right over there. And I'll just have to draw it perfectly to scale but this makes sense, that this right over here would be the midpoint.