
In this unit, we're going to
look at how to add and subtract

complex numbers. Now when you're
at school, you first learn to

add and subtract using the
counting numbers. That's numbers

1234 and so on.

And every time you meet a new
set of numbers, you learn a new

rule or a new process for how
you can add and subtract them.

So for example, when you meet
fractions, you learn it to add 2

fractions, you must write them
both over a common denominator.

No, it's the same with complex
numbers. You just need to learn

the correct process to how to
add and subtract them, and with

complex numbers is quite
straightforward. It's based on

the principle you have seen
before in algebra where you

collect together some like
terms, so we're going to do is

going to start by adding two
algebraic expressions and then

see how that carries over into
adding together 2 complex

numbers. So let's take two
algebraic expressions. Let's

take the expressions for plus.

70 and 2
+ 3 two.

So we have two expressions are
going to do is going to add

these two things together, so
we're going to have 4 + 70.

Plus 2 + 3 T.

Now, the principle of collecting
together, like turns, simply

says looking your long
expression and look for terms

that are somehow the same. So in
this expression we have the four

on the two, which is just
numbers. So we can put those two

together. So 4 + 2 gives us 6.

And we have these two terms
which have both got Tees in

them, so we've got plus 70 + 3
more Tees so that gives

altogether plus 10 T.

So we started with two algebraic
expressions. We've added them

up. And we simplified by
collecting together terms that

are the same terms that are just
numbers and terms that have got

teasing them. Now adding
together complex numbers works

in exactly the same sort of way,
so we're going to do now is take

two complex numbers and add them
together. So I'm going to take

the complex #4 + 7 I.

And the complex number 2 + 3 I.

So this is first complex
number and this is my second

complex number. I'm going to
add them together.

Now what I do is I say well.

If if I take away the brackets,
I won't have changed anything.

Now I can look at this
expiration, which doesn't have

any brackets in and I can look
for terms that I can collect

together so I can collect the
four in the two together to get

6, and I can collect the plus
Seven I and the plus three I

together to get plus 10I.

And so that's the answer. When I
add these two complex numbers

together, I get this new complex
number 6 + 10 I and you'll see

that in fact all we've done is
we've added together the real

parts of the two complex
numbers, and we've added

together the imaginary parts of
the two complex numbers to get

the answer. So let's do
another example. OK, in this

example will just take two
different complex numbers and

add them together.

So we'll take the complex
numbers 5 + 6 I.

And the complex number 7  3 I.

Will add those two complex

numbers together. So just as
before will write everything out

without any brackets.

And then we look for the terms
that we can collect together so

we can collect together the real
parts. That's just the numbers.

The real numbers 5 + 7 to give

us 12. And then we can collect
together the two imaginary parts

plus six I minus three I giving
us plus 3I.

And so the answer when we had
these two complex numbers

together is 12 + 3 I and once
again we see that we've done is.

We've just added together the
real parts and we've added

together the imaginary parts.

Now subtraction works in
exactly the same way, we just

have to be careful that we
make sure that we do take away

the whole of the second
complex number. 'cause

sometimes people might just
take away the real part and

forget to take away the
imaginary part as well. So

let's see how we can do that.

So if we go back to our first
pair of numbers 4 + 7 I.

We're going to take away from
that 2 + 3 I.

So when we remove the brackets,
this time we have to be careful

to make the minus operate on the
whole of this complex number.

So from the first complex number
we have 4 + 7 I.

But then when we take these
brackets out, we get minus 2.

Then we get minus plus three
eyes are getting minus three I.

And now we're in a position to
collect together like terms.

Just as we've been doing in all
the examples so far.

So our numbers are 4  2 which

is 2. And then our terms
with I in them are plus

Seven. I minus three I,
so that gives us plus 4I.

And so the answer when we do
this subtraction 4 + 7 I take

away 2 + 3. I gives us a
complex number 2 + 4 I and

again we can see that what
we've done is. We know this,

I'm subtracting the real parts
4 takeaway two to give us two

and subtracted the imaginary
part 7 takeaway three to give

us four and that goes with the
eye because that's the

imaginary part.

The one last example now just to
make sure we've got this well

and truly understood, we're
going to take start with a

complex number 5 + 6 I.

I'm going to take away from that
the complex number 7  3 I.

Let me just repeat the process.
We've seen several Times Now we

remove the brackets to start off

with. So nothing happens to
the first 2 terms, but the

minus operates on everything
that's in this next set of

brackets. So we get minus 7.

Then we get minus minus three.
I2 minus is making a plus.

Three, I and now we collect
together the real numbers we

collect together the imaginary

numbers. We have 5  7 which
is minus two. We have plus 6I

plus three I which is +9 I.

And so that's our answer. This
time and again we see that what

we've done is, we've subtracted
the real parts 5  7 gives us

minus two, and then we've
subtracted the imaginary parts 6

minus minus three is 6 + 3,
which is 9 because there's the

imaginary parts. That's nine I
in the answer.

So the way we add or subtract
2 complex numbers is simply to

use algebraic manipulation and
collect together like terms in

the next unit. We look at how
to multiply 2 complex numbers

together.