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In this unit, we're going to
look at the complex conjugate.

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Every complex number as
associated with it, another

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complex number, which is called
its complex conjugate.

4
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And you find the complex
conjugate of a complex number

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simply by changing the imaginary
part of that number.

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This is best illustrated by
looking at some examples.

7
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So here in this table we've got
three different complex numbers,

8
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and we're going to do is going
to find the complex conjugate of

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each of these three numbers.

10
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So we start by looking at the
complex #4 + 7 I.

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On the way to find the complex
conjugate is to change the sign

12
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of the imaginary part. So that
means that the plus sign changes

13
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to a minus sign, so the complex
conjugate is 4 minus.

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Seven I.

15
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Here's another complex number 1
- 3. I defined its complex

16
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number. We change the sign of
the imaginary part. In other

17
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words, we change this minus sign
to a plus. So we get the complex

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number 1 + 3 I.

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As another complex number minus
4 - 3 I.

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And defined its complex
conjugate. Again we change the

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sign of the imaginary part. We
don't need to be worried about

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what the sign of the real part
is. We just changing the sign of

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the imaginary part and so we get
minus 4 + 3 I.

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So whenever we start with any
complex number, we can find

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its complex conjugate very
easily. We just change the

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sign of the imaginary
partners.

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Now the complex conjugate has a
very special property and we'll

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see what that is by doing an

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example. OK, what we're going to
do is we're going to take a

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complex #4 + 7 I I'm going to
multiply it by its own complex

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conjugate, which is 4 - 7 I, and
we're going to see what we get.

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So we do. 4 * 4 is
16 four times minus Seven. I is

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minus 28 I.

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Plus Seven I times four is

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plus 28I. And plus Seven
I minus Seven I is minus

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49 I squared.

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Now when we come to tidy this

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up. The 16 stays there.

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We have minus 28I Plus 28I, so
they cancel each other out, so

40
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we're left with no eyes.

41
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So there's nothing coming from
those two terms, and from this

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term on the end, we've got minus
49. I squared. We remember that

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I squared is minus one, so we
got minus 49 times minus one, so

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that's plus 49.

45
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And 16 +
49 is 65.

46
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So when we multiply the two
complex numbers together 4 + 7 I

47
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and its complex conjugate 4 - 7
I we find that the answer we get

48
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is 65. There was the answer is a
purely real number, it has no

49
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imaginary part or an imaginary
part of 0.

50
00:03:28,230 --> 00:03:32,118
That is quite important. So two
complex numbers multiplying

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together to give a real number.

52
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Let's see if it's always
happens. Let's try another pair

53
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and complex number and its
complex conjugate and see what

54
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happens then. OK, in this
example we're just going to take

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another complex number and its
complex conjugate and multiply

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them together. So what we've got
is 1 - 3 I. Its complex

57
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conjugate is 1 + 3 I let's
multiply them together. 1 * 1 is

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one. One times plus three. I
is plus 3I.

59
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Minus three items, one is minus

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three I. And minus three I times
plus three I is minus 9.

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I squat.

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Always do now is tidy this up.
That means we combined together

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are terms in I and we use the
fact that I squared is equal to

64
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minus one. So we get one start
plus three. I minus three I, so

65
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that's no eyes and then minus
nine isquared. Remembering that

66
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I squared is minus one, we've
got minus nine times minus one,

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giving is plus 9, which is an
answer of text.

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So once again we've
multiplied complex number by

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its complex conjugate and
we've got a real number.

70
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Now this is a very important
property and it doesn't just

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happen in the two examples that
I've picked, it happens that

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every complex number. If you
pick any complex, then be like

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and multiply it by its complex
conjugate, you will get a real

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number and that turns out to be
very important when we come to

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learn how to divide complex
numbers, which is what will be

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doing in the next unit.