www.mathcentre.ac.uk/.../06-Complex-ConjugateF61Mb.mp4

• 0:03 - 0:06
In this unit, we're going to
look at the complex conjugate.
• 0:07 - 0:10
Every complex number as
associated with it, another
• 0:10 - 0:13
complex number, which is called
its complex conjugate.
• 0:14 - 0:17
And you find the complex
conjugate of a complex number
• 0:17 - 0:20
simply by changing the imaginary
part of that number.
• 0:21 - 0:24
This is best illustrated by
looking at some examples.
• 0:25 - 0:29
So here in this table we've got
three different complex numbers,
• 0:29 - 0:33
and we're going to do is going
to find the complex conjugate of
• 0:33 - 0:35
each of these three numbers.
• 0:35 - 0:40
So we start by looking at the
complex #4 + 7 I.
• 0:41 - 0:46
On the way to find the complex
conjugate is to change the sign
• 0:46 - 0:50
of the imaginary part. So that
means that the plus sign changes
• 0:50 - 0:54
to a minus sign, so the complex
conjugate is 4 minus.
• 0:55 - 0:56
Seven I.
• 0:57 - 1:02
Here's another complex number 1
- 3. I defined its complex
• 1:02 - 1:06
number. We change the sign of
the imaginary part. In other
• 1:06 - 1:12
words, we change this minus sign
to a plus. So we get the complex
• 1:12 - 1:14
number 1 + 3 I.
• 1:16 - 1:20
As another complex number minus
4 - 3 I.
• 1:20 - 1:24
And defined its complex
conjugate. Again we change the
• 1:24 - 1:28
sign of the imaginary part. We
don't need to be worried about
• 1:28 - 1:33
what the sign of the real part
is. We just changing the sign of
• 1:33 - 1:37
the imaginary part and so we get
minus 4 + 3 I.
• 1:38 - 1:42
complex number, we can find
• 1:42 - 1:46
its complex conjugate very
easily. We just change the
• 1:46 - 1:48
sign of the imaginary
partners.
• 1:49 - 1:53
Now the complex conjugate has a
very special property and we'll
• 1:53 - 1:55
see what that is by doing an
• 1:55 - 2:00
example. OK, what we're going to
do is we're going to take a
• 2:00 - 2:05
complex #4 + 7 I I'm going to
multiply it by its own complex
• 2:05 - 2:10
conjugate, which is 4 - 7 I, and
we're going to see what we get.
• 2:10 - 2:18
So we do. 4 * 4 is
16 four times minus Seven. I is
• 2:18 - 2:19
minus 28 I.
• 2:21 - 2:24
Plus Seven I times four is
• 2:24 - 2:31
plus 28I. And plus Seven
I minus Seven I is minus
• 2:31 - 2:33
49 I squared.
• 2:35 - 2:36
Now when we come to tidy this
• 2:36 - 2:39
up. The 16 stays there.
• 2:40 - 2:45
We have minus 28I Plus 28I, so
they cancel each other out, so
• 2:45 - 2:47
we're left with no eyes.
• 2:48 - 2:52
So there's nothing coming from
those two terms, and from this
• 2:52 - 2:57
term on the end, we've got minus
49. I squared. We remember that
• 2:57 - 3:02
I squared is minus one, so we
got minus 49 times minus one, so
• 3:02 - 3:03
that's plus 49.
• 3:03 - 3:08
And 16 +
49 is 65.
• 3:09 - 3:14
So when we multiply the two
complex numbers together 4 + 7 I
• 3:14 - 3:19
and its complex conjugate 4 - 7
I we find that the answer we get
• 3:19 - 3:24
is 65. There was the answer is a
purely real number, it has no
• 3:24 - 3:27
imaginary part or an imaginary
part of 0.
• 3:28 - 3:32
That is quite important. So two
complex numbers multiplying
• 3:32 - 3:35
together to give a real number.
• 3:35 - 3:39
Let's see if it's always
happens. Let's try another pair
• 3:39 - 3:42
and complex number and its
complex conjugate and see what
• 3:42 - 3:46
happens then. OK, in this
example we're just going to take
• 3:46 - 3:49
another complex number and its
complex conjugate and multiply
• 3:49 - 3:55
them together. So what we've got
is 1 - 3 I. Its complex
• 3:55 - 4:01
conjugate is 1 + 3 I let's
multiply them together. 1 * 1 is
• 4:01 - 4:06
one. One times plus three. I
is plus 3I.
• 4:07 - 4:09
Minus three items, one is minus
• 4:09 - 4:16
three I. And minus three I times
plus three I is minus 9.
• 4:16 - 4:18
I squat.
• 4:19 - 4:24
Always do now is tidy this up.
That means we combined together
• 4:24 - 4:29
are terms in I and we use the
fact that I squared is equal to
• 4:29 - 4:35
minus one. So we get one start
plus three. I minus three I, so
• 4:35 - 4:39
that's no eyes and then minus
nine isquared. Remembering that
• 4:39 - 4:43
I squared is minus one, we've
got minus nine times minus one,
• 4:43 - 4:47
giving is plus 9, which is an
• 4:48 - 4:51
So once again we've
multiplied complex number by
• 4:51 - 4:55
its complex conjugate and
we've got a real number.
• 4:56 - 5:00
Now this is a very important
property and it doesn't just
• 5:00 - 5:03
happen in the two examples that
I've picked, it happens that
• 5:03 - 5:06
every complex number. If you
pick any complex, then be like
• 5:06 - 5:10
and multiply it by its complex
conjugate, you will get a real
• 5:10 - 5:14
number and that turns out to be
very important when we come to
• 5:14 - 5:17
learn how to divide complex
numbers, which is what will be
• 5:17 - 5:19
doing in the next unit.
Title:
www.mathcentre.ac.uk/.../06-Complex-ConjugateF61Mb.mp4
Video Language:
English
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../06-Complex-ConjugateF61Mb.mp4