L7 1 2Review Polar And Rectangular Forms

>> So complex numbers consist of two
parts: A real part and an imaginary part,

and we typically write them then as z a
complex number is equal to a plus j B.

Because there's two parts it requires

a complete plane to
represent them graphically.

A real number we can show
on the real number line,

an imaginary number we can show
along an imaginary number line.

But when you've got
both the real and imaginary part,

we form what is known as
the complex plane where

the horizontal axis of the plane
is referred to as the real axis,

and we plot the real value of the
complex number along the horizontal axis.

The vertical axis becomes
the imaginary axis

where we plot the imaginary part
of our complex number.

So, for example here we have
z is equal to two plus j.

We come over one,

two units along the real part because
in this case a the real part is two,

and we go up one unit along

the imaginary axis because
our imaginary part is one times j.

So, here we'd have one j,

two j, three j.

You can think of J almost as

a unit vector or a quantity
showing us which axis we're on.

The distance along this axis is two.

the distance along this axis
is one times j. All right.

We here are representing

this real number or

this complex number in what is
known as rectangular coordinates,

a horizontal value and a vertical value.

As you know we can also represent points in

a plane using polar coordinates where

the two coordinate values are

a angle of rotation away from
the horizontal axis which we call theta,

and the distance from the origin
out to the point along

that ray is known as the magnitude
of the complex number.

So, we can write z in polar form,

in terms of this magnitude and an angle.

Now, how did the polar and rectangular
coordinates relate to each other?

We can go from rectangular to
polar using Pythagorean's theorem.

So, from rectangular to polar we
can calculate the magnitude of z.

That's simply equal to,

using Pythagorean's theorem, the
square root of the two sides.

Well this is A, so it would be
the square root of A squared,

and the distance along
the imaginary axis is b.

So, A squared plus B squared and

theta the angle is equal
to the inverse tangent

of the opposite over the adjacent or

the inverse of the imaginary over the real

or the inverse tangent of b over a.

Now, to go from polar coordinates
back to rectangular coordinates,

we again use the right
triangle and recognize

that the distance along
the horizontal axis is A,

that's simply equal to the projection
of that point down onto

the real axis which of course is

the magnitude of z times
the cosine of the angle.

So, A then, is equal to

the magnitude of z times
the cosine of the angle theta,

and b, the imaginary component is equal

to the magnitude of z
times the sine of theta.

Again, is just for
the projection of this point,

onto the imaginary axis.

So, that takes us from polar coordinates,

back to rectangular coordinates.

Now let's just do an example real
quick to show how this works.

In fact let's do it for
this quantity right there.

The complex number z equals two plus j
given to us in rectangular coordinates,

how do we get the polar
representation of it?

While coming down here,

the magnitude of z is going to equal

the square root of the real part
squared which is two squared,

plus the imaginary part.

The imaginary part is
coming up here one unit,

again we don't square the j.

The distance there, for
the geometry in this circuit,

in this problem, the distance
there is one unit,

so it'll be one squared and that
equals the square root of five.

Theta is equal to the arc-tangent
of the imaginary part which is one,

it's not J, it's one.

Again, this distance is one unit.

So, the one imaginary part
over the real part two,

and that turns out to be 26.6 degrees.

Now to show how we take it and go back.

First of all, this number
then in polar coordinates,

we would represent as square root
of five angle 26.6 degrees.

Okay? Now, how do we take

this number in polar form and go
back into rectangular coordinates?

Well, it tells us right here that A,

the real part of our number
is equal to A is

equal to the magnitude of Z which we
found to be the square root of five,

times the cosine of the angle which
is in this case is 26.6 degrees.

When you're doing this make sure
you got your calculator is set in

degrees and not radians if
you're working in degrees.

That then of course is going to equal A,

the distance along
the real axis which we know

and when you can't do that calculation
if you find that is equal to two,

and b the imaginary part then
is equal to the square root of

five times the sine of 26.6
degrees which equals one.