>> 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.