WEBVTT

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In this unit, we're going to
look at how to divide 2 complex

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numbers. Now, division of
complex numbers is rather more

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complicated than addition,
Subtraction, and multiplication.

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And Division of complex numbers
relies on two very important

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principles. The first is that
when you take a complex number

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and multiply by its complex
conjugate, you get a real

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number. The second important
principle is that when you have

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a fraction, you can multiply the
numerator and the denominator.

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That's the number on the top on
the number on the bottom of the

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fraction by the same value, and
not change the value of a

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fraction. So for example, if
you start with a fraction of

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half and you multiply the
top and bottom by 5, you get

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5/10 and the value of five
10s is the same as the value

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of 1/2. And that's really
going to be very important

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when we come into being able
to workout. How to divide 1

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complex number by another.
So let's look at an example.

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So we're going to take the
complex #4 + 7 I. I'm going to

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divide it by the complex number
1 - 3. I now remember the

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division is the same thing. It's
a fraction, so this complex

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number divided by this one. We
can just write a Swan complex

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number over another complex

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number. So now we have a
fraction we can say is that we

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won't change the value of this
fraction if we multiply the

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numerator and the denominator by
the same value.

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I'm going to choose to multiply
the denominator by 1 + 3 I.

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1 + 3 I is the complex conjugate
of 1 - 3 I and we choose this

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complex conjugate so that when
we do the multiplication, what's

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in the denominator will turn out
to be a real number.

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So for multiplying the
denominator by 1 + 3 I we've got

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to multiply the numerator by 1 +

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3 I. So that way we have
multiplied the numerator and

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denominator by the same value,
so we haven't changed the value

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of the answer.

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So let's now multiply these two
fractions together. We multiply

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out the two terms in the
numerator. We multiply out the

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two terms in the denominator, so
we get 4 * 1 is 4.

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4 * 3 I is 12 I.

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Seven 8 * 1 is 7 I.

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+78 times plus three. I is plus
21 by squares.

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So that's multiplied. The two
terms in the numerator. Now we

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multiply the two terms in the
denominator to get 1 * 1 one

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times plus 3I.

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Minus three items one.

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And minus three I times
plus three. I give this

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minus nine I squared.

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What time do this up?

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21 I squared is 21 times minus
one, so that's minus 21, so

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we've got 4 - 21 is minus 17.

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12 + 7 I is 99, so we've
got plus 99.

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And then in the denominator.

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I squared is minus one, so we've
got minus nine times minus one

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is plus nine, 1 + 9 is 10.

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And three I minus three. I is
nothing. So the management turns

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disappear. So we've ended up
with a real denominator so we

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could leave our answer like
this. Or we could split it up as

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minus 17 over 10 + 19 over 10

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I. And if we want we
could write as minus one point 7

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+ 10.9 I.

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So that's our answer. When we
divide 4 + 7, I buy 1 - 3.

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I get minus one point 7 + 1.9.

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Now let's do another example to
illustrate the principals again.

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Here are two more complex
numbers 2 - 5 I and minus 4 + 3

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i's going to divide the first
one by the second one.

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And we write those as a fraction
2 - 5 I over minus 4 +

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3. I now the way to do it
is to multiply. Want to multiply

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the denominator by its complex
conjugate, which is minus 4 - 3

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I. And because we're multiplying
the denominator by this value,

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we must multiply the numerator.

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By this value as well.

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Now we multiply out the
numerator and denominator.

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So we have two times minus four
is minus 8 two times minus

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three. I is minus six I.

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Minus 5I Times minus four is
plus 20I and minus 5I times

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minus three. I is plus 15
I squared.

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And then in the dominator we
have minus four times minus 4

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inches 16. Minus four
times minus three I,

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which is plus 12 I.

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Plus three I times minus 4
inches minus 12 I.

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I'm plus three I times
minus three I, which is

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minus nine. I squared.

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And now he tidies up.

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59 squared is minus 15, so we've
got minus 8 - 15 is minus 23.

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Minus six I plus
20I is plus 49.

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So that's the numerator
simplified, and then the

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denominator. We've got minus
nine. I squared, so that's plus

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nine. We got 16 + 9 is
25 and 12. I minus 12. I

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that disappears, leaving us with
a real denominator, which is

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what we wanted. So we can write
that as minus 23 over 25 +

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14 over 25 I.

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Which we could also write
us minus N .92 +

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.56 high.

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And so that's the result
of doing this division.

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Now in the next unit, we'll
look at something called the

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organ diagram, which is a way
of graphically representing

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complex numbers.