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# Intro to complex number polar form

Video transcript

We're asked to adjust the angle
in radius of the orange plotted complex number-- so that's
this right over here-- to match the blue plotted
complex number negative 3.5 plus 6.06i. So this is this one
right over here. So just to get our bearings,
this vertical axis, this is the imaginary axis. This horizontal axis
is the real axis. The real part of this
number is negative 3.5, and so we've plotted that on
the real axis-- negative 1, negative 2, negative 3.5. And then the imaginary
part is 6.06. So we go up 1, 2, 3, 4, 5,
6, and then a very little bit to get to 6.06. So they've plotted it like that. And now they want
us to think about it in kind of the polar form is
one way of thinking about it. And so they tell us that we
can adjust the angle, which is often called the argument
when you're plotting up a complex number in this form,
and the radius, which is often called the modulus. So let's change that. So let's see, if we
increase the radius, so then the thing is going
to go out a little bit. If we increase the
angle, that's pi over 3. This is pi over 2. So we increase the
angle to 2 pi over 3. And this seems to be
the right direction, and we just have to increase
the radius a little bit. So there we have a radius
of 7, an angle of 2/3 pi. Now, let's verify for ourselves
that this complex number and this complex number
really are the same thing. We've already seen it right over
here in this actual diagram. We've plotted it
both in I guess you could say, rectangular
form and polar form, although this is a complex
axis, and this is a real axis. So let me copy and paste the
information that we have here. So let me just copy
and paste all of that. And then I can put it on
my little scratch pad. So let me throw it
right over there, and let's actually verify that. So here we have an argument, or
an angle I guess I could say. So this angle, often
referred to as phi when you're dealing
in the complex plane, this is equal to 2/3 pi. And this is, of
course, in radians. And the modulus, which
is really the distance from the origin to the complex
number, this right over here, by just manipulating that little
tool, we saw that to be 7. So let's verify that this is
really the same complex number as negative 3.5 plus 6.06i. Well, we can
represent this as 7-- this is the modulus,
or the radius. 7 times e to the 2/3 pi i, which
we know from Euler's identity is the exact same thing. This is equal to-- and this
is one of really the most mind-blowing things
in all of mathematics. This is the same thing as
7 times cosine of 2/3 pi plus i sine of 2/3 pi. And they already wrote that
down for us right over there. Now let's verify what
cosine of 2/3 pi is and what sine of 2/3 pi is. So if this is 2/3 as
an angle-- and now you can almost even
think of the unit circle. So this is 2/3 pi. So actually, just let me just
draw a unit circle like this because I don't want
us to confuse just our traditional
rectangular coordinates with what we're doing
on the polar axis. So this isn't a
polar-- we're not graphing imaginary
numbers right over here. This is just the
traditional unit circle. This is our x-axis
this is our y-axis. 2/3 pi is the same
thing as 120 degrees. So it looks just like what
we did over here, so 2/3 pi. So let me draw it as neatly as
I can, which isn't that neatly. So that angle is 2/3 pi. Our cosine of 2/3 pi is going
to be our x-coordinate here. And sine of 2/3 pi is going to
be our y-coordinates over here. So what's our x-coordinate? Well, if this right over
here is 120 degrees, then this is 60 degrees. This is a 30-60-90 triangle. And we know this
is a unit circle. The radius is 1. So the 30-degree side
will have length 1/2. So this coordinate
right over here is going to be negative 1/2. And then the side opposite
the 60-degree side is square root of
3 times the 1/2. So this right over here is going
to be square root of 3 over 2. And you could verify that on
a calculator, if you like. So cosine of 2/3
pi is negative 1/2. So this is going to be equal
to 7 times negative 1/2. That's cosine of 2/3 pi
plus i times the square root of 3 over 2. So let's verify
that that's actually the same thing as
that over there. Well, 7 times negative 1/2
is negative 3.5 plus i. I'll just write it lik e7
square root of 3 over 2i. So we already see that the real
parts are definitely the same. It's negative 3.5. And I'll just get
a calculator out to evaluate this and see
if it is roughly 6.06. So let me get my
trusty TI-85 out. And then I want to evaluate
7 times the square root of 3, and then that divided
by 2 is indeed 6.06, if we round to the
nearest hundredths. So this is approximately equal
to negative 3.5 plus 6.06i, which is exactly the complex
number that we started with.