L7 1 5Example Complex Calculations

>> 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.