>> Let's take a look at how we do adding, subtracting, multiplying,
and dividing complex numbers and see
how different forms of the complex number lends
themselves to easier algebraic or arithmetic computations.
We have two different complex numbers here Z1,
which is equal to four plus J3 in rectangular coordinates.
In its polar form,
Z1 is also equal to five angle 37.87 degrees.
Of course, in it's complex exponential form which is still polar,
it still has the polar values magnitude
five angle e to the j angle 37.87 degrees.
Z2 equal to 2 minus J5,
in polar form it's that and it's complex exponential form is that.
Let's just plot these out right here,
tag it in just so that we can see what we have here.
In the complex plane we're going over 4 and up 3.
So it's about like that.
On this one, we're coming over two and going down five.
So maybe it's down here something like that for Z2 and Z1. All right.
Let's start this off by just adding Z1 plus Z2.
Z1 plus Z2, addition is most easily done in rectangular coordinates.
Because in that you simply add the real parts and the imaginary parts.
So Z1 plus Z2 will be 4 plus J3 plus Z2,
which is 2 minus J5.
When you do that, you get, let's see,
4 plus 2 is 6 and J3 minus J5 is a minus J2.
Pretty easy. All right.
How about multiplying.
Let's do Z1 times Z2.
Well, before we go on, let me just say that multiplying or
adding in polar form really isn't very,
polar form isn't conducive to adding unless you want to do it graphically.
Of course, adding graphically you take Z1 and Z2 and you tip to tail them.
So if that's Z2,
then you'd add in Z1 like that and the resulting complex number would be that.
Generally speaking, we don't add in polar form
unless you're using your calculator and your calculator can do all things.
So now, let's look at multiplying Z1 times Z2.
We get then Z1 is
4 plus J3 times Z2 which is 2 minus J5.
We see here that we have two binomials multiplied together.
That means we need to foil them.
Distribute both of these terms to both of those terms.
When you do that you get 4 times 2 is 8,
4 times a minus J5 is a minus J20.
J3 times 2 is a plus J6.
J3 times minus J5,
we've go to be careful here.
3 times 5 is 15.
But we've go to be careful with the sign plus times a minus is minus.
But J times J is a negative one.
So we have a minus times minus becomes plus.
So we have two imaginary terms,
two real terms.8 plus 15 is 23.
The negative J20 plus J6 is a minus J14.
So multiplying two complex numbers in rectangular form requires
us to foil them.
I'll leave it to you and you stop
the video right here and go ahead and convert
this rectangular complex number into its polar form.
Let me just give the answer. The answer is 26.93 angle negative 31.33.
All right. Now, let's multiply Z1 times Z2.
Only doing it in polar coordinates using our polar form of these.
In polar we have
Z1 times Z2 that's equal to 5.
I'm going to write it using its complex exponential form.
5e to the J37.87 times
Z2 which is 5.39 angle negative,
I'm doing an exponential form,
e to the negative J68.2.
Of course, these two forms are equivalent forms.
It's just that it's more obvious what we're
doing when we write them as exponentials.
So when you do that 5 times
5.39 is equal to 26.93.
Now we have e to the J positive 37.887 and we have e to the minus J68.2.
So when we add the exponents,
the negative 68.2 plus the J37.87 gives us an angle of,
so the minus J angle.
31.33.
What we see is we get the same answer
foiling and then converting to polar as we
did by just starting out in polar form and multiplying
the coefficients and adding the exponents.
I'm going to leave it to you to take and convert
this expression here in
polar back to rectangular
and convince yourself that you get the same thing there.
Go and stop the video and do that now. All right.
Now, we've got addition,
by the way, I should have mentioned it,
subtraction is just the same as addition or when you're
subtracting the real parts and then the subtracting the imaginary parts.
We've now done multiplication in both rectangular form and in polar form.
Now, let's look at at dividing two complex numbers.
Let's do Z1 divided by Z2.
You'll recall from your college algebra that
this gets to be a little bit ugly and then at least a little involved.
We've got one in doing this in rectangular form.
We have 4 plus J3 divided by 2 minus J5.
You'll recall that in college algebra,
they taught us to do this division by multiplying numerator and
denominator by the complex conjugate of the denominator.
Effectively rationalizing the denominator so
that we get a pure real number in the denominator,
and then the numerator falls wherever it may.
So we're going to multiply numerator and denominator
by 2 plus J5 over 2 plus J5.
2 plus J5 is the conjugate of 2 minus J5.
Then you can go through and do the foiling because
we have this complex number times this complex number,
it's got to be foiled.
I'm going to leave that to you to show that it turns out to
be negative 7 plus J26 divided by,
now let me show the details down here just to remind you what happens here.
We've got 2 times 2 is 4.
We have 2 times a positive J5,
that's a positive J10.
Now we have a negative J5 times 2,
that gives me a minus J10.
Then we have a negative J5 times a positive J5.
Again, we've got to be careful with the signs here.
5 times 5 is 25.
J times J is a negative 1 minus times a plus is a minus.
So we've got a minus from the signs a minus from
the J squared that gives us a positive.
So this then turns out to be negative 7 plus J26 divided by 29,
which works out to be equal to negative 0.24 plus J0.9.
So the division of Z1 divided by Z2 in
rectangular coordinates gives us this rectangular coordinate.
Now, hub goodness, let's go ahead and convert this to polar coordinates.
In polar coordinates, we're going to have the magnitude of that as equal to
the square root of negative 0.24 squared plus 0.9 squared.
That turns out to be 0.928.
That's the magnitude.
Now, the angle here gets to be a little bit tricky.
The angle is going to be the arc tangent
of 0.9 divided by negative 0.24.
Now, the arc tangent button on your calculator returns a value between
plus or minus Pi halves or plus or minus 90 degrees.
This becomes ambiguous.
To find out what the actual angle is,
you take the arc tangent of 0.9 divided by negative 0.24
and you come up with a negative 75 degrees.
So this turns out to be
0.928 angle negative 75 degrees.
But that's not exactly right.
Coming back here and looking at our complex number,
we see that we are in not the fourth quadrant,
negative 75 it run out of room,
let's just draw it down here.
It has this down here at negative 75 degrees.
But when we look at this number here,
we're at negative 0.24.
The real part is negative,
so we're over here and up 0.9.
In reality where we're at with this complex number
is 180 degrees off from this negative 75 degrees.
So the actual angle is negative 75 plus 180.
This angle right here is actually 105 degrees.
So when you're doing it in
rectangular coordinates using your arc tangent button
on your calculator you got be
careful and look at the actual coordinates that you're working with,
because your arc tangent button can cause you some grief.
Let's do this calculation directly from polar form.
We have as Z1 over Z2 is equal to,
in polar form Z1 is 5e to the J37.87 divided by Z2,
which is 5.39e to the minus J68.2.
Now we can do that directly and 5 divided by 5.39,
that is infact equal to 0.928e to the,
now the exponent is going to be J,
and we've got 37.87 minus 68.82,
that turns out to be the 105 and some round off error there.
But it's positive J105 degrees.
When you do it this way,
already in the polar form,
the ambiguity that the arc tangent button
introduces is we avoid that ambiguity.