### Video Transcript

In this lesson, we will learn how
to add and subtract complex numbers. We’ll begin by recapping what we
mean by a complex number and what it means for two complex numbers to be equal. We will then learn how to add and
subtract these numbers, extending this idea into solving simple equations involving
complex numbers.

Remember, a complex number 𝑧 is a
number of the form 𝑎 plus 𝑏𝑖. It is important that 𝑎 and 𝑏 are
both real numbers. And 𝑖 is defined as the solution
to the equation 𝑥 squared equals negative one. We say that 𝑖 squared is equal to
negative one and sometimes 𝑖 is equal to the squared root of negative one.

And for our complex number 𝑎 plus
𝑏𝑖, we say that the real part of 𝑧 is 𝑎 and the imaginary part is 𝑏. It’s important that the imaginary
part is 𝑏, not 𝑏𝑖. It’s essentially the coefficient of
𝑖. And just as the set of real numbers
is denoted by the letter ℝ, the set of complex numbers is denoted by the letter ℂ,
as shown.

Before we can perform addition and
subtraction and indeed solve equations with complex numbers, we should define what
it means for two complex numbers to be equal. We already saw that a complex
number is made up of two parts: a real part and an imaginary part. 𝑎 and 𝑏 are the real and
imaginary parts, respectively. And they are both part of the real
number set.

Let’s say then we have two complex
numbers 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖. We want to know what it means for
these two complex numbers to be equal. Well, in fact, it follows that
their real parts must be equal and their imaginary parts must separately be
equal. We can say then that 𝑎 plus 𝑏𝑖
equals 𝑐 plus 𝑑𝑖 if 𝑎 is equal to 𝑐 and 𝑏 is equal to 𝑑. In other words, two complex numbers
are equal if their real parts are equal and separately their imaginary parts are
equal. And of course, the equivalent is
also true. If 𝑎 is equal to 𝑐 and 𝑏 is
equal to 𝑑, for two complex numbers 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖, then 𝑎 plus
𝑏𝑖 must be equal to 𝑐 plus 𝑑𝑖. Let’s consider our problem for
which this definition might be useful.

If the complex numbers seven plus
𝑎𝑖 and 𝑏 minus three 𝑖 are equal, what are the values of 𝑎 and 𝑏?

Remember, for two complex numbers
to be equal, their real parts must be equal and their imaginary parts must also be
equal. And the beauty of this fact is it
takes a problem about complex numbers and makes it purely about real numbers, since
both the real parts and imaginary parts of each complex number must be real
numbers.

Let’s have a look at the complex
numbers seven plus 𝑎𝑖 and 𝑏 minus three 𝑖 then. The real part of the first complex
number is seven, and the real part of our second complex number is 𝑏. The imaginary part of our first
complex number is 𝑎, and the imaginary part of our second complex number is
negative three. It follows then that seven must be
equal to 𝑏 and 𝑎 must be equal to negative three. Both negative three and seven are
real numbers, which satisfies our criteria for the real and imaginary parts of a
complex number. So for the complex numbers seven
plus 𝑎𝑖 and 𝑏 minus three 𝑖 to be equal, 𝑎 must be equal to negative three and
𝑏 must be equal to seven.

And what about adding and
subtracting complex numbers? Remember, a complex number is the
result of adding a real number and an imaginary number. We can compare this a little to the
idea of an algebraic expression like four plus seven 𝑥. This is the result of adding a
number and a term in 𝑥. We could add, for example, four
plus seven 𝑥 and another expression such as two plus five 𝑥 by individually adding
the numbers to get six and adding the terms in 𝑥. That’s seven 𝑥 and five 𝑥, which
is 12𝑥.

We can add complex numbers in
exactly the same way, remembering that the letter 𝑖 doesn’t actually represent a
variable. But it’s the solution to the
equation 𝑥 squared equals negative one. Let’s generalise this for complex
numbers 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖. Their sum is 𝑎 plus 𝑏𝑖 plus 𝑐
plus 𝑑𝑖. And we can add the real parts 𝑎
and 𝑐, and we get 𝑎 plus 𝑐, and then add their imaginary parts. That’s 𝑏 plus 𝑑. So we see that the sum of these two
complex numbers is 𝑎 plus 𝑐 plus 𝑏 plus 𝑑 𝑖.

Their difference is 𝑎 plus 𝑏𝑖
minus 𝑐 plus 𝑑𝑖. This time, we add their real parts
and we get 𝑎 minus 𝑐. And we add their imaginary
parts. We get 𝑏. And then we distribute the brackets
and we get negative 𝑑. So the difference is 𝑎 minus 𝑐
plus 𝑏 minus 𝑑 𝑖. So to add and subtract complex
numbers, we add or subtract their real parts and separately add or subtract their
imaginary parts.

In fact, what we’ve seen so far is
that we can take a problem about complex numbers and turn it into a problem about
real numbers by considering the real and imaginary parts. This is great because it means we
can take the skills we already have for working with real numbers and extend them
into working with complex numbers. Let’s see what this might look
like.

What is negative nine plus seven
plus four 𝑖 plus negative four minus four 𝑖 minus one plus three 𝑖?

Remember, we can add or subtract
complex numbers by adding their real parts and separately adding their imaginary
parts. Here we have four complex
numbers. Now it might not look like it, but
we could say that negative nine is actually a complex number. It’s negative nine plus zero
𝑖.

So to solve this problem, we’re
going to work out the real part first. That’s negative nine plus seven
minus four minus one, which is negative seven. Similarly, the imaginary parts are
zero, four, negative four, and negative three.

Remember, we distribute this final
set of parentheses. And a negative multiplied by a
positive is a negative. That gives us negative three. So the real part of our solution is
negative seven, and the imaginary part is negative three. So negative nine plus seven plus
four 𝑖 plus negative four minus four 𝑖 minus one plus three 𝑖 is negative seven
minus three 𝑖.

Now we also know that many
algebraic properties can be extended into the concept of negative numbers. We could actually have collected
like terms. Distributing that final set of
parentheses and then rearranging slightly, we get negative nine plus seven plus
negative four minus one plus four 𝑖 plus negative four 𝑖 minus three 𝑖, which
once again gives us negative seven minus three 𝑖. This latter method, the one of
collecting like terms, is generally the one that we use when adding and subtracting
complex numbers.

It’s useful to remember though that
there are other methods that can work. Let’s see why.

If 𝑟 equals five plus two 𝑖 and
𝑠 equals nine minus 𝑖, find the real part of 𝑟 minus 𝑠.

Here we have two complex numbers to
find as five plus two 𝑖 and nine minus 𝑖. We can see that the real part of 𝑟
is five and the real part of 𝑠 is nine. The imaginary part of 𝑟 is two and
the imaginary part of 𝑠 is negative one. We’re being asked to find the real
part of the difference between 𝑟 and 𝑠. And we could absolutely work out
the entire answer to 𝑟 minus 𝑠 by collecting like terms. That’s five plus two 𝑖 minus nine
minus 𝑖.

It’s important that we use these
parentheses here because it reminds us that we’re subtracting everything inside
these brackets, nine minus 𝑖. If we distribute these parentheses,
we get five plus two 𝑖 minus nine plus 𝑖, since subtracting a negative is the same
as adding a positive. Then we would simplify by
collecting like terms. However, that’s probably a little
more work than we really need to do.

In fact, we recall that, to
subtract complex numbers, we simply subtract the real parts and then subtract the
imaginary parts separately. We’re being asked to work out the
real parts of the complex number 𝑟 minus 𝑠. So actually, we just need to
subtract the real part of 𝑠 from the real part of 𝑟. We can formalise this and say that
the real part of 𝑟 minus 𝑠 equals the real part of 𝑟 minus the real part of
𝑠. We already saw that the real part
of 𝑟 is five and the real part of 𝑠 is nine. Five minus nine is negative
four. So the real part of 𝑟 minus 𝑠 in
this case is negative four.

Now that we’ve established what it
means for two complex numbers to be equal and learned how to add and subtract
complex numbers, this will allow us to solve simple equations involving these types
of numbers.

Determine the real numbers 𝑥 and
𝑦 that satisfy the equation five 𝑥 plus two plus three 𝑦 minus five 𝑖 equals
negative three plus four 𝑖.

Let’s look carefully at what we’ve
been given. We have been given two complex
numbers that we’re told are equal to each other. Now I know it doesn’t look like it,
but that expression to the left of the equal sign is indeed a complex number. Remember, a complex number is one
of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. And we’re told that 𝑥 and ~~𝑖~~ [𝑦] are
real numbers. And this means that the expression
five 𝑥 plus two must be real and three 𝑦 minus five must be real. So five 𝑥 plus two plus three 𝑦
minus five 𝑖 is a complex number. It has a real part of five 𝑥 plus
two and an imaginary part of three 𝑦 minus five.

Next, we’ll recall what it actually
means for two complex numbers to be equal. We see that two complex numbers 𝑎
plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 are equal if 𝑎 is equal to 𝑐 and 𝑏 is equal to 𝑑. In other words, their real parts
must be equal and their imaginary parts must separately be equal.

Let’s begin with the real parts in
our question. We saw that the real part of the
complex number on the left is five 𝑥 plus two. And on the right, it’s negative
three. This means that five 𝑥 plus two
must be equal to negative three. We’ll solve this as normal by
applying a series of inverse operations. We’ll subtract two from both sides,
and then we’ll divide through by five. And we see that 𝑥 is equal to
negative one.

Let’s repeat this process for the
imaginary parts. We said that the imaginary part for
our number on the left is three 𝑦 minus five. And on the right, we can see it’s
four. This means that three 𝑦 minus five
must be equal to four. We can add five to both sides of
this equation. And then we’ll divide through by
three. And we see that 𝑦 must be equal to
three. And we’ve solved the equation for
𝑥 and 𝑦. 𝑥 equals negative one and 𝑦
equals three.

In fact, it’s always sensible to
check our answers by substituting them back into the equation and making sure that
it makes sense. If we do, we get five multiplied by
negative one plus two plus three multiplied by three minus five 𝑖. This does indeed give us negative
three plus four 𝑖 as required.

Our final example uses everything
we’ve looked at in this video, with just a little more complexity.

Let 𝑧 one equal four 𝑥 plus two
𝑦𝑖 and 𝑧 two equal four 𝑦 plus 𝑥𝑖, where 𝑥 and 𝑦 are real numbers. Given that 𝑧 one minus 𝑧 two is
equal to five plus two 𝑖, find 𝑧 one and 𝑧 two.

Let’s look carefully at what we’ve
been given. We’ve been given two complex
numbers in terms of 𝑥 and 𝑦. And we know these are complex
numbers because we’re told that 𝑥 and 𝑦 are real numbers. That’s an important definition of a
complex number. Both the real and imaginary parts
of the complex numbers must be made up by real numbers. We’re also told that the difference
between these two numbers is five plus two 𝑖.

Let’s recall: to subtract complex
numbers, we simply subtract the real parts and then subtract the imaginary parts
separately. This means that the real part of 𝑧
one minus 𝑧 two must be equal to the difference between the real parts of 𝑧 one
and 𝑧 two. The real part of 𝑧 one minus 𝑧
two is five. The real part of 𝑧 one is four 𝑥,
and the real part of our second complex number is four 𝑦. So five is equal to four 𝑥 minus
four 𝑦.

Let’s repeat this process for our
imaginary numbers. The imaginary part of the
difference is two. The imaginary part of 𝑧 one is two
𝑦, and the imaginary part of 𝑧 two is 𝑥. So two equals two 𝑦 minus 𝑥. And now we see we have a pair of
simultaneous equations in 𝑥 and 𝑦. We can use any method we’re
comfortable with to solve these.

Now I think substitution lends
itself quite nicely to these equations. Let’s rearrange this second
equation to make 𝑥 the subject. We add 𝑥 to both sides and then
subtract two. And we get 𝑥 equals two 𝑦 minus
two. We then substitute this into our
first equation. And we see that five is equal to
four lots of our value of 𝑥, which is two 𝑦 minus two. And then we subtract that four
𝑦.

We distribute these parentheses by
multiplying each term by four. And we see that five is equal to
eight 𝑦 minus eight minus four 𝑦. Eight 𝑦 minus four 𝑦 is four
𝑦. We’ll solve this equation by adding
eight to both sides to get 13 equals four 𝑦. And then we’ll divide through by
four. And we see that 𝑦 is equal to 13
over four.

We could substitute this value back
into any of our original equations. But it’s sensible to choose the
rearranged form of the second equation. 𝑥 is equal to two multiplied by 13
over four minus two. Two multiplied by 13 over four is
the same as 13 over two. And two is the same as four over
two. 13 over two minus four over two is
nine over two. And this is usually where we would
stop.

But we’ve been asked to find the
complex numbers 𝑧 one and 𝑧 two. So we need to substitute our values
for 𝑥 and 𝑦 into each of these. We get 𝑧 one equals four
multiplied by nine over two plus two multiplied by 13 over four 𝑖. That’s 18 plus 13 over two 𝑖. 𝑧 two is four multiplied by 13
over four plus nine over two 𝑖. This is 13 plus nine over two
𝑖.

And it’s sensible to check our
answer by subtracting 𝑧 two from 𝑧 one and checking we do indeed get five plus two
𝑖. We subtract their real parts. 18 minus 13 is five, as
required. And we subtract their imaginary
parts. 13 over two minus nine over two is
four over two, which simplifies to two. And the imaginary part is two as
required.

In this video, we’ve seen that we
can turn a problem about complex numbers into one involving real numbers by
considering their real and imaginary parts. And we’ve seen that this can be
useful because we know how to add, subtract, and equate with real numbers
already. We’ve also learned that we can
extend ideas about rules for algebraic expressions to help us work with complex
numbers.

We’ve seen that two complex numbers
are equal if individually their real components are equal and their imaginary parts
are equal. And finally, we’ve learned that we
can add and subtract complex numbers by adding or subtracting their real parts and
adding or subtracting their imaginary parts.