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

Every complex number as
associated with it, another

complex number, which is called
its complex conjugate.

And you find the complex
conjugate of a complex number

simply by changing the imaginary
part of that number.

This is best illustrated by
looking at some examples.

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

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

each of these three numbers.

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

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

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

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

Seven I.

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

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

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

number 1 + 3 I.

As another complex number minus
4  3 I.

And defined its complex
conjugate. Again we change the

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

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

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

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

its complex conjugate very
easily. We just change the

sign of the imaginary
partners.

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

see what that is by doing an

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

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

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

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

minus 28 I.

Plus Seven I times four is

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

49 I squared.

Now when we come to tidy this

up. The 16 stays there.

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

we're left with no eyes.

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

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

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

that's plus 49.

And 16 +
49 is 65.

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

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

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

imaginary part or an imaginary
part of 0.

That is quite important. So two
complex numbers multiplying

together to give a real number.

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

and complex number and its
complex conjugate and see what

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

another complex number and its
complex conjugate and multiply

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

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

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

Minus three items, one is minus

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

I squat.

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

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

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

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

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

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

So once again we've
multiplied complex number by

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

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

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

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

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

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

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

doing in the next unit.