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.