Distance & midpoint of complex numbers

so we have two complex numbers here the complex number Z is equal to 2 plus 3i and the complex number W is equal to negative 5 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 their 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 plane 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 going to have go as high as positive 2 and the real axis and as low as negative 5 along the real axis let's go 1 2 3 4 5 1 2 3 4 5 and along the imaginary axis we go as high as positive 3 and as low as negative 1 so we could do 1 2 3 and we could do 1 2 3 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 2 and then we and then we have 3 times I so the imaginary part is 3 so we would go right over here so this is 2 and this is 3 right over here 2 plus 3i so that right over there is Z now let's plot W W is negative 5 1 2 3 4 5 negative 5 minus I so this is negative 1 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 of this line right over here and to figure that out we can really just think about the Pythagorean theorem when if you say hey what about the distance formula and not icon in the 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 could 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 go it we're going from negative five along the real axis to two what is two minus negative five well it's seven if we go five to get to to get to the zero along the real axis and then we go two more to get to two so the length of this right over here seven and what is the length of 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 could 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 forty-nine plus sixteen and what is that going to be equal to that is 65 so X that's right yeah fifty-nine plus another six is 65 X is equal to the square root of sixty-five and let's see 65 you can't factor this you 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 as 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 are some complex number let's just call it a is the midpoint its 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 1 3 minus one minus one over two times I and this is equal to let's see 2 plus negative five is negative three so this is negative three halves plus this is three minus one is negative is negative two over two is so let's see three mining make sure I'm doing this right three so I'm taking the mean three minus one is two divided by two is one so three plus three negative three halves plus I is the midpoint between those two and if we plot it we can verify that that actually makes sense so real part negative three halves 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 obviously I didn't draw it perfectly to scale but this makes sense that this right over here would be the midpoint