WEBVTT

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>> Let's take a look at how we do
adding, subtracting, multiplying,

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and dividing complex numbers and see

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how different forms of
the complex number lends

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themselves to easier algebraic
or arithmetic computations.

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We have two different
complex numbers here Z1,

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which is equal to four plus J3
in rectangular coordinates.

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In its polar form,

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Z1 is also equal to
five angle 37.87 degrees.

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Of course, in it's complex exponential
form which is still polar,

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it still has the polar values magnitude

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five angle e to the j angle 37.87 degrees.

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Z2 equal to 2 minus J5,

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in polar form it's that and it's
complex exponential form is that.

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Let's just plot these out right here,

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tag it in just so that we
can see what we have here.

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In the complex plane we're
going over 4 and up 3.

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So it's about like that.

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On this one, we're coming
over two and going down five.

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So maybe it's down here something
like that for Z2 and Z1. All right.

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Let's start this off by
just adding Z1 plus Z2.

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Z1 plus Z2, addition is most easily
done in rectangular coordinates.

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Because in that you simply add
the real parts and the imaginary parts.

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So Z1 plus Z2 will be 4 plus J3 plus Z2,

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which is 2 minus J5.

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When you do that, you get, let's see,

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4 plus 2 is 6 and J3
minus J5 is a minus J2.

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Pretty easy. All right.

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How about multiplying.

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Let's do Z1 times Z2.

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Well, before we go on, let me
just say that multiplying or

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adding in polar form really isn't very,

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polar form isn't conducive to adding
unless you want to do it graphically.

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Of course, adding graphically you take
Z1 and Z2 and you tip to tail them.

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So if that's Z2,

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then you'd add in Z1 like that and the
resulting complex number would be that.

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Generally speaking, we
don't add in polar form

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unless you're using your calculator
and your calculator can do all things.

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So now, let's look at
multiplying Z1 times Z2.

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We get then Z1 is

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4 plus J3 times Z2 which is 2 minus J5.

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We see here that we have
two binomials multiplied together.

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That means we need to foil them.

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Distribute both of these terms
to both of those terms.

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When you do that you get 4 times 2 is 8,

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4 times a minus J5 is a minus J20.

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J3 times 2 is a plus J6.

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J3 times minus J5,

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we've go to be careful here.

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3 times 5 is 15.

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But we've go to be careful with
the sign plus times a minus is minus.

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But J times J is a negative one.

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So we have a minus times
minus becomes plus.

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So we have two imaginary terms,

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two real terms.8 plus 15 is 23.

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The negative J20 plus J6 is a minus J14.

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So multiplying two complex numbers
in rectangular form requires

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us to foil them.

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I'll leave it to you and you stop

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the video right here and
go ahead and convert

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this rectangular complex number
into its polar form.

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Let me just give the answer. The answer
is 26.93 angle negative 31.33.

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All right. Now, let's multiply Z1 times Z2.

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Only doing it in polar coordinates
using our polar form of these.

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In polar we have

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Z1 times Z2 that's equal to 5.

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I'm going to write it using
its complex exponential form.

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5e to the J37.87 times

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Z2 which is 5.39 angle negative,

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I'm doing an exponential form,

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e to the negative J68.2.

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Of course, these two forms
are equivalent forms.

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It's just that it's more obvious what we're

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doing when we write them as exponentials.

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So when you do that 5 times

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5.39 is equal to 26.93.

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Now we have e to the J positive 37.887
and we have e to the minus J68.2.

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So when we add the exponents,

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the negative 68.2 plus the
J37.87 gives us an angle of,

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so the minus J angle.

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31.33.

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What we see is we get the same answer

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foiling and then converting to polar as we

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did by just starting out in
polar form and multiplying

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the coefficients and adding the exponents.

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I'm going to leave it to
you to take and convert

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this expression here in

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polar back to rectangular

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and convince yourself that
you get the same thing there.

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Go and stop the video and
do that now. All right.

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Now, we've got addition,

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by the way, I should have mentioned it,

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subtraction is just the same
as addition or when you're

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subtracting the real parts and then
the subtracting the imaginary parts.

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We've now done multiplication in
both rectangular form and in polar form.

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Now, let's look at at
dividing two complex numbers.

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Let's do Z1 divided by Z2.

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You'll recall from your
college algebra that

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this gets to be a little bit ugly
and then at least a little involved.

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We've got one in doing
this in rectangular form.

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We have 4 plus J3 divided by 2 minus J5.

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You'll recall that in college algebra,

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they taught us to do this division
by multiplying numerator and

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denominator by the complex conjugate
of the denominator.

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Effectively rationalizing
the denominator so

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that we get a pure real number
in the denominator,

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and then the numerator
falls wherever it may.

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So we're going to multiply
numerator and denominator

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by 2 plus J5 over 2 plus J5.

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2 plus J5 is the conjugate of 2 minus J5.

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Then you can go through
and do the foiling because

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we have this complex number times
this complex number,

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it's got to be foiled.

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I'm going to leave that to you
to show that it turns out to

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be negative 7 plus J26 divided by,

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now let me show the details down here
just to remind you what happens here.

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We've got 2 times 2 is 4.

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We have 2 times a positive J5,

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that's a positive J10.

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Now we have a negative J5 times 2,

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that gives me a minus J10.

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Then we have a negative
J5 times a positive J5.

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Again, we've got to be
careful with the signs here.

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5 times 5 is 25.

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J times J is a negative 1
minus times a plus is a minus.

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So we've got a minus from
the signs a minus from

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the J squared that gives us a positive.

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So this then turns out to be
negative 7 plus J26 divided by 29,

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which works out to be equal
to negative 0.24 plus J0.9.

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So the division of Z1 divided by Z2 in

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rectangular coordinates gives
us this rectangular coordinate.

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Now, hub goodness, let's go ahead and
convert this to polar coordinates.

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In polar coordinates, we're going to
have the magnitude of that as equal to

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the square root of negative
0.24 squared plus 0.9 squared.

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That turns out to be 0.928.

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That's the magnitude.

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Now, the angle here gets
to be a little bit tricky.

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The angle is going to be the arc tangent

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of 0.9 divided by negative 0.24.

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Now, the arc tangent button on
your calculator returns a value between

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plus or minus Pi halves or
plus or minus 90 degrees.

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This becomes ambiguous.

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To find out what the actual angle is,

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you take the arc tangent of
0.9 divided by negative 0.24

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and you come up with a negative 75 degrees.

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So this turns out to be

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0.928 angle negative 75 degrees.

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But that's not exactly right.

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Coming back here and looking
at our complex number,

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we see that we are in
not the fourth quadrant,

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negative 75 it run out of room,

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let's just draw it down here.

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It has this down here
at negative 75 degrees.

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But when we look at this number here,

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we're at negative 0.24.

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The real part is negative,

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so we're over here and up 0.9.

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In reality where we're at
with this complex number

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is 180 degrees off from
this negative 75 degrees.

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So the actual angle is
negative 75 plus 180.

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This angle right here is
actually 105 degrees.

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So when you're doing it in

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rectangular coordinates using
your arc tangent button

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on your calculator you got be

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careful and look at the actual
coordinates that you're working with,

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because your arc tangent button
can cause you some grief.

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Let's do this calculation
directly from polar form.

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We have as Z1 over Z2 is equal to,

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in polar form Z1 is 5e to
the J37.87 divided by Z2,

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which is 5.39e to the minus J68.2.

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Now we can do that directly
and 5 divided by 5.39,

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that is infact equal to 0.928e to the,

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now the exponent is going to be J,

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and we've got 37.87 minus 68.82,

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that turns out to be the 105
and some round off error there.

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But it's positive J105 degrees.

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When you do it this way,

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already in the polar form,

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the ambiguity that the arc tangent button

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introduces is we avoid that ambiguity.