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.