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## Topic B: Lessons 9-12: Distance and midpoint of complex numbers

# Distance & midpoint of complex numbers

CCSS.Math:

## 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.