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

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

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

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1234 and so on.

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

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

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

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

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

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

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complex numbers is quite
straightforward. It's based on

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the principle you have seen
before in algebra where you

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

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going to start by adding two
algebraic expressions and then

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see how that carries over into
adding together 2 complex

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

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take the expressions for plus.

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70 and 2
+ 3 two.

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

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

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Plus 2 + 3 T.

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

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says looking your long
expression and look for terms

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

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

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together. So 4 + 2 gives us 6.

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

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

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altogether plus 10 T.

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

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up. And we simplified by
collecting together terms that

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

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teasing them. Now adding
together complex numbers works

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

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

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the complex #4 + 7 I.

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And the complex number 2 + 3 I.

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So this is first complex
number and this is my second

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

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Now what I do is I say well.

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

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

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

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

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

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together to get plus 10I.

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

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

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

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

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together the imaginary parts of
the two complex numbers to get

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

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example will just take two
different complex numbers and

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add them together.

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

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And the complex number 7 - 3 I.

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Will add those two complex

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numbers together. So just as
before will write everything out

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without any brackets.

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

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

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The real numbers 5 + 7 to give

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

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

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And so the answer when we had
these two complex numbers

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

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

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together the imaginary parts.

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Now subtraction works in
exactly the same way, we just

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

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the whole of the second
complex number. 'cause

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sometimes people might just
take away the real part and

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forget to take away the
imaginary part as well. So

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let's see how we can do that.

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

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

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

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

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

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

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

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

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

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So our numbers are 4 - 2 which

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

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

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

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

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

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

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and subtracted the imaginary
part 7 takeaway three to give

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

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imaginary part.

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

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

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complex number 5 + 6 I.

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

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

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remove the brackets to start off

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

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

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brackets. So we get minus 7.

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

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

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collect together the imaginary

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

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plus three I which is +9 I.

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

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

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

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

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

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

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use algebraic manipulation and
collect together like terms in

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

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together.