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.