### Video Transcript

In this video, we will learn how to
use the conjugate of a complex number to evaluate expressions. We will begin by defining what we
mean by the complex conjugate, before considering their properties and how these can
be exploited to help us solve equations involving complex numbers. Throughout this video, we will look
to, where possible, derive general results that can be used in more complicated
complex number problems.

Every single complex number has
associated with it another complex number, known as its conjugate. The definition of the word
conjugate is having features in common but opposite or inverse in some
particular. Outside of mathematics, it can mean
to juxtapose or join together, indicating that a complex number and its conjugate
have a special relationship. Let’s look at the definition. For a complex number of the form 𝑎
plus 𝑏𝑖, its conjugate denoted by 𝑧 bar or 𝑧 star is 𝑎 minus 𝑏𝑖. In layman’s terms, the conjugate of
a complex number is found by changing the sign for the imaginary part of the
number.

For example, a complex number given
as three plus two 𝑖 — we’ll have a complex conjugate of three minus two 𝑖. Similarly, a complex number, four
minus six 𝑖 — we’ll have a conjugate of four plus six 𝑖. And in fact, the complex conjugate
of the conjugate is four minus six 𝑖. And we can see that we can
generalize this and say that the complex conjugate of the conjugate is simply the
original number. It’s 𝑧.

And what about a purely real
number? Will this have a conjugate? Well yes. In fact, we can say that a real
number is of the form 𝑎. This is actually a complex
number. But it’s one of the form 𝑎 plus
zero 𝑖. Its imaginary part is zero. Since we change the sign of the
imaginary part to find the conjugate, the conjugate of this number will be 𝑎 minus
zero 𝑖. But of course, this is still
𝑎. So the conjugate of the real number
is simply that number. Another beauty of the conjugate is
that it shares all of the same properties as any other complex number. It’s distributive over addition and
multiplication. And we’re going to see now what
that might look like.

If 𝑠 equals eight plus two 𝑖,
what is 𝑠 plus 𝑠 star?

Remember, 𝑠 star is the conjugate
of the complex number 𝑠, given by eight plus two 𝑖. We can say that the complex
conjugate of a number 𝑧, given by 𝑎 plus 𝑏𝑖, is 𝑧 star equals 𝑎 minus
𝑏𝑖. Our complex number has a real part
of eight and an imaginary part of two. This means its complex conjugate is
eight minus two 𝑖. And this also means that the sum of
the two numbers is eight plus two 𝑖 plus eight minus two 𝑖. And we add these by adding their
real parts and separately adding their imaginary parts, which we often think of like
collecting like terms. Eight plus eight is 16. And two 𝑖 minus two 𝑖 is
zero. And we see that 𝑠 plus 𝑠 star is
16.

Notice how the sum of a complex
number and its conjugate is just a real number. And this makes a lot of sense if we
consider the general form. 𝑧 plus 𝑧 star is 𝑎 plus 𝑏𝑖
plus 𝑎 minus 𝑏𝑖. 𝑏𝑖 minus 𝑏 𝑖 is zero. And we therefore see that the sum
of a complex number with its complex conjugate is simply two 𝑎. Or alternatively, we might say that
the sum of a complex number and its conjugate is two lots of the real part of that
number.

Similarly, their difference is 𝑎
plus 𝑏𝑖 minus 𝑎 minus 𝑏𝑖. We distribute the second bracket by
multiplying each part by negative one. And we get 𝑎 plus 𝑏𝑖 minus 𝑎
plus 𝑏𝑖. This time, 𝑎 minus 𝑎 is zero,
which is equal to two 𝑏𝑖. We can therefore say that the
difference between a complex number and its conjugate is two 𝑖 multiplied by the
imaginary part of that complex number.

We’ll now look at a detailed
example of an equation involving the sum and difference of a complex number and its
conjugate.

Find the complex number 𝑧 which
satisfies the following equations. 𝑧 plus 𝑧 star is equal to
negative five. 𝑧 star minus 𝑧 is equal to three
𝑖.

Remember, 𝑧 star represents the
conjugate of the complex number 𝑧. 𝑧 will be of the form 𝑎 plus
𝑏𝑖, where 𝑎 and 𝑏 are real numbers. And 𝑧 star will be of the form 𝑎
minus 𝑏𝑖. We changed the sign of the
imaginary part. The first equation tells us the sum
of these two numbers. We can form an expression for their
sum by using the general form of the complex number. It’s 𝑎 plus 𝑏𝑖 plus 𝑎 minus
𝑏𝑖. This simplifies to two 𝑎 or two
multiplied by the real part of 𝑧.

Now in fact, we know that the sum
of these numbers is negative five. So we can say that negative five is
equal to two multiplied by the real part of 𝑧. And we solve by dividing through by
two. And we see that the real part of 𝑧
is equal to negative five over two.

The second equation tells us the
difference of these two numbers. That’s 𝑎 minus 𝑏𝑖 minus 𝑎 plus
𝑏𝑖. We distribute the second bracket by
multiplying each term by negative one. And we get 𝑎 minus 𝑏𝑖 minus 𝑎
minus 𝑏𝑖, which is negative two 𝑏𝑖. This is equal to negative two 𝑖
multiplied by the imaginary part of 𝑧. And of course, we know that this is
actually equal to three 𝑖. So we see that three 𝑖 is equal to
negative two 𝑖 multiplied by the imaginary part of 𝑧. To solve this equation, we divide
through by negative two 𝑖. And since 𝑖 divided by 𝑖 is one,
we see that the imaginary part of 𝑧 is negative three over two.

It’s useful to spot that we could
alternatively have multiplied through by negative one. That would’ve given us 𝑧 minus 𝑧
star is equal to negative three 𝑖. But this would’ve resulted in the
same solution. So we know that the complex number
𝑧, which satisfies the two equations given, has a real part of negative five over
two and an imaginary part of negative three over two. So this solution is negative five
over two minus three over two 𝑖.

You should now be able to see why
learning the definition of the sum and difference of a complex number and its
conjugate can be a real time saver. Next, we’ll consider the product of
a complex number and its conjugate.

Find the complex conjugate of one
plus 𝑖 and the product of this number with its complex conjugate.

Remember, the complex conjugate is
found by changing the sign of the imaginary part of the complex number. This means that the complex
conjugate of one plus 𝑖 is one minus 𝑖. And we want to find the product of
one plus 𝑖 and one minus 𝑖. We find the product of these two
numbers just as we would with any two binomials.

The FOIL method can be a nice way
to do this. F stands for first. We multiply the first term in the
first bracket by the first term in the second bracket. One multiplied by one is one. O stands for outer. We multiply the outer two
terms. One multiplied by negative 𝑖 is
negative 𝑖. I stands for inner. We multiply the inner terms. And 𝑖 multiplied by one is 𝑖. And finally, L stands for last. We multiply the last term in each
bracket. 𝑖 multiplied by negative 𝑖 is
negative 𝑖 squared. And of course, 𝑖 squared is equal
to negative one. Since negative 𝑖 plus 𝑖 is zero,
this becomes one minus negative one, which is two. And the product of this number with
its complex conjugate is two.

We can generalize this result. And we’ll soon see that there are a
number of applications for the complex conjugate.

Let 𝑎 plus 𝑏𝑖 be a complex
number whose conjugate is 𝑎 minus 𝑏𝑖. Their product is 𝑎 plus 𝑏𝑖
multiplied by 𝑎 minus 𝑏𝑖. If we expand these brackets as
before, we get 𝑎 squared minus 𝑎𝑏𝑖 plus 𝑎𝑏𝑖 minus 𝑏 squared 𝑖 squared. And of course, negative 𝑎𝑏𝑖 plus
𝑎𝑏𝑖 is zero. And you might notice this is just
like the difference of two squares. We’ll substitute negative one for
𝑖 squared. And we see that the product of this
complex number with its conjugate is 𝑎 squared plus 𝑏 squared. So 𝑧 multiplied by 𝑧 star is 𝑎
squared plus 𝑏 squared. For our last two examples, we’ll
consider some more complicated questions involving sums and products of complex
numbers and their conjugates.

Consider 𝑧 equals five minus 𝑖
root three and 𝑤 equals root two plus 𝑖 root five. Part one, calculate 𝑧 star and 𝑤
star. Part two, find 𝑧 star plus 𝑤 star
and 𝑧 plus 𝑤 star. Part three, find 𝑧 star 𝑤 star
and 𝑧𝑤 star.

In this question, we’ve been given
two complex numbers. And we need to find their
conjugates. Remember, to find the conjugate of
a complex number, we change the sign of its imaginary part. This means that the complex
conjugate of five minus 𝑖 root three is five plus 𝑖 root three. Now, don’t worry that the 𝑖 is in
front of the root three here. Yes, that’s not of the general
form. But it’s a sensible way to write it
when we’re dealing with roots.

If we, instead, chose to write root
three multiplied by 𝑖, it could look a little bit like we’re finding the square
root of three 𝑖 rather than the square root of three multiplied by 𝑖. Next, we see since 𝑤 is root two
plus 𝑖 root five, its conjugate is root two minus 𝑖 root five.

For part two, we need to calculate
two numbers. We need to find the sum of the
conjugates. And we need to find the conjugate
of the sum of the original complex numbers. Let’s begin by finding the sum of
their conjugates. That’s five plus 𝑖 root three plus
root two minus 𝑖 root five. We can find their sum by collecting
like terms. And when we do, we see that 𝑧 star
plus 𝑤 star is five plus root two plus 𝑖 multiplied by root three minus root
five. And we can also work out 𝑧 plus 𝑤
star.

This time, we add 𝑧 and 𝑤 first
before finding the conjugate. That’s the conjugate of five minus
𝑖 root three plus root two plus 𝑖 root five. Once again, we do this by adding
the real parts and then the imaginary parts. We get five plus root two plus 𝑖
multiplied by negative root three plus root five. So we change the sign of the
imaginary part to find the conjugate. And we get five plus root two minus
𝑖 multiplied by negative root three plus root five. And in fact, if we multiply the
imaginary part by negative one, we see that this is equal to five plus root two plus
𝑖 multiplied by root three minus root five. Notice that 𝑧 star plus 𝑤 star is
actually the same as 𝑧 plus 𝑤 star.

For this third part, we need to
find the product of the conjugate of 𝑧 and 𝑤. That’s five plus 𝑖 root three
multiplied by root two minus 𝑖 root five. Multiplying the first term in each
bracket, we get five root two. Multiplying the outer terms, we get
negative five 𝑖 root five. Multiplying the inner terms, that’s
𝑖 root three multiplied by root two, gives us 𝑖 root six. And multiplying the last terms
gives us 𝑖 root three multiplied by negative 𝑖 root five, which is negative 𝑖
squared multiplied by root 15. And since 𝑖 squared is equal to
negative one, this last term becomes positive root 15. Collecting like terms, we see that
the product of the conjugate of 𝑧 and 𝑤 is five root two plus root 15 plus 𝑖
multiplied by root six minus five root five.

Next, we find the product of 𝑧 and
𝑤 and then find their conjugate. This time, their product is five
minus 𝑖 root three multiplied by root two plus 𝑖 root five. Expanding these brackets, and we
get five root two plus root 15 minus 𝑖 multiplied by root six minus five root
five. And it follows that the conjugate
of this number is five root two plus root 15 plus 𝑖 multiplied by root six minus
five root five, once again.

We’ve seen in this example that,
for complex numbers 𝑧 and 𝑤, the sum of their conjugates is equal to the conjugate
of their sum. And we’ve also seen that the
product of their conjugates is equal to the conjugate of their product. And this in fact is a general rule,
which holds for all complex numbers.

Solve two 𝑧 minus 𝑧 bar equals
five in ℂ.

Here we have a complex number. And we can say that that could be
of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. 𝑧 bar is its conjugate. That’s 𝑎 minus 𝑏𝑖. And ℂ is used to denote the set of
complex numbers. Let’s substitute 𝑧 and 𝑧 bar into
our equation.

When we do, we see that two
multiplied by 𝑎 plus 𝑏𝑖 minus 𝑎 minus 𝑏𝑖 equals five. Then, we distribute the brackets by
multiplying the real and imaginary part by the number on the outside. For the first bracket, that’s two
multiplied by 𝑎 and two multiplied by 𝑏𝑖. And for the second bracket, that’s
negative one multiplied by 𝑎 plus negative one multiplied by negative 𝑏𝑖. So we get two 𝑎 plus two 𝑏𝑖
minus 𝑎 plus 𝑏𝑖 equals five. And of course, we can collect like
terms. And we see that 𝑎 plus three 𝑏𝑖
is equal to five.

Now, this number is purely
real. Or we can say it’s a complex number
whose imaginary part is zero. And once we’ve identified that, we
can equate the real and imaginary parts. We see that 𝑎 must be equal to
five. And three 𝑏 must be equal to
zero. In fact, if three 𝑏 is equal to
zero, 𝑏 must also be equal to zero. We’re solving for 𝑧. And we’ve established that 𝑎 — its
real part is equal to five. And 𝑏 — its imaginary part is
equal to zero. So we could say that 𝑧 is equal to
five plus zero 𝑖 though we don’t need to write the imaginary part. So we say that 𝑧 is simply equal
to five.

In this video, we’ve learned that a
complex number of the form 𝑎 plus 𝑏𝑖 has a complex conjugate 𝑎 minus 𝑏𝑖. And this is often denoted by 𝑧
star or sometimes 𝑧 bar. We’ve also seen that, for two
complex numbers 𝑧 one and 𝑧 two, there exist a whole set of rules that relate the
complex numbers with their conjugates. And finally, we’ve learned that a
complex number is equal to its conjugate if and only if its imaginary part is zero,
in other words, if it’s a real number.