• 0:03 - 0:07
In this unit, we're going to
look at how to add and subtract
• 0:07 - 0:11
complex numbers. Now when you're
at school, you first learn to
• 0:11 - 0:15
counting numbers. That's numbers
• 0:15 - 0:16
1234 and so on.
• 0:17 - 0:21
And every time you meet a new
set of numbers, you learn a new
• 0:21 - 0:24
rule or a new process for how
you can add and subtract them.
• 0:24 - 0:28
So for example, when you meet
fractions, you learn it to add 2
• 0:28 - 0:31
fractions, you must write them
both over a common denominator.
• 0:31 - 0:36
No, it's the same with complex
numbers. You just need to learn
• 0:36 - 0:40
the correct process to how to
add and subtract them, and with
• 0:40 - 0:43
complex numbers is quite
straightforward. It's based on
• 0:43 - 0:46
the principle you have seen
before in algebra where you
• 0:46 - 0:50
collect together some like
terms, so we're going to do is
• 0:50 - 0:53
going to start by adding two
algebraic expressions and then
• 0:53 - 0:57
see how that carries over into
• 0:57 - 1:00
numbers. So let's take two
algebraic expressions. Let's
• 1:00 - 1:01
take the expressions for plus.
• 1:01 - 1:08
70 and 2
+ 3 two.
• 1:08 - 1:13
So we have two expressions are
going to do is going to add
• 1:13 - 1:18
these two things together, so
we're going to have 4 + 70.
• 1:19 - 1:23
Plus 2 + 3 T.
• 1:24 - 1:28
Now, the principle of collecting
together, like turns, simply
• 1:28 - 1:31
expression and look for terms
• 1:31 - 1:36
that are somehow the same. So in
this expression we have the four
• 1:36 - 1:41
on the two, which is just
numbers. So we can put those two
• 1:41 - 1:44
together. So 4 + 2 gives us 6.
• 1:45 - 1:50
And we have these two terms
which have both got Tees in
• 1:50 - 1:55
them, so we've got plus 70 + 3
more Tees so that gives
• 1:55 - 1:57
altogether plus 10 T.
• 1:58 - 2:03
So we started with two algebraic
• 2:03 - 2:07
up. And we simplified by
collecting together terms that
• 2:07 - 2:12
are the same terms that are just
numbers and terms that have got
• 2:12 - 2:15
together complex numbers works
• 2:15 - 2:21
in exactly the same sort of way,
so we're going to do now is take
• 2:21 - 2:26
two complex numbers and add them
together. So I'm going to take
• 2:26 - 2:28
the complex #4 + 7 I.
• 2:29 - 2:33
And the complex number 2 + 3 I.
• 2:34 - 2:39
So this is first complex
number and this is my second
• 2:39 - 2:42
complex number. I'm going to
• 2:43 - 2:45
Now what I do is I say well.
• 2:46 - 2:50
If if I take away the brackets,
I won't have changed anything.
• 2:56 - 3:00
Now I can look at this
expiration, which doesn't have
• 3:00 - 3:04
any brackets in and I can look
for terms that I can collect
• 3:04 - 3:08
together so I can collect the
four in the two together to get
• 3:08 - 3:13
6, and I can collect the plus
Seven I and the plus three I
• 3:13 - 3:14
together to get plus 10I.
• 3:17 - 3:21
And so that's the answer. When I
• 3:21 - 3:26
together, I get this new complex
number 6 + 10 I and you'll see
• 3:26 - 3:30
that in fact all we've done is
• 3:30 - 3:33
parts of the two complex
• 3:33 - 3:37
together the imaginary parts of
the two complex numbers to get
• 3:37 - 3:41
another example. OK, in this
• 3:41 - 3:44
example will just take two
different complex numbers and
• 3:44 - 3:45
• 3:47 - 3:50
So we'll take the complex
numbers 5 + 6 I.
• 3:52 - 3:55
And the complex number 7 - 3 I.
• 3:56 - 3:58
• 3:58 - 4:04
numbers together. So just as
before will write everything out
• 4:04 - 4:05
without any brackets.
• 4:09 - 4:13
And then we look for the terms
that we can collect together so
• 4:13 - 4:16
we can collect together the real
parts. That's just the numbers.
• 4:16 - 4:19
The real numbers 5 + 7 to give
• 4:19 - 4:24
us 12. And then we can collect
together the two imaginary parts
• 4:24 - 4:28
plus six I minus three I giving
us plus 3I.
• 4:31 - 4:35
these two complex numbers
• 4:35 - 4:40
together is 12 + 3 I and once
again we see that we've done is.
• 4:40 - 4:43
• 4:43 - 4:44
together the imaginary parts.
• 4:46 - 4:49
Now subtraction works in
exactly the same way, we just
• 4:49 - 4:53
have to be careful that we
make sure that we do take away
• 4:53 - 4:55
the whole of the second
complex number. 'cause
• 4:55 - 4:58
sometimes people might just
take away the real part and
• 4:58 - 5:01
forget to take away the
imaginary part as well. So
• 5:01 - 5:03
let's see how we can do that.
• 5:04 - 5:10
So if we go back to our first
pair of numbers 4 + 7 I.
• 5:11 - 5:16
We're going to take away from
that 2 + 3 I.
• 5:18 - 5:22
So when we remove the brackets,
this time we have to be careful
• 5:22 - 5:26
to make the minus operate on the
whole of this complex number.
• 5:27 - 5:31
So from the first complex number
we have 4 + 7 I.
• 5:32 - 5:36
But then when we take these
brackets out, we get minus 2.
• 5:38 - 5:42
Then we get minus plus three
eyes are getting minus three I.
• 5:44 - 5:47
And now we're in a position to
collect together like terms.
• 5:47 - 5:49
Just as we've been doing in all
the examples so far.
• 5:51 - 5:55
So our numbers are 4 - 2 which
• 5:55 - 6:00
is 2. And then our terms
with I in them are plus
• 6:00 - 6:05
Seven. I minus three I,
so that gives us plus 4I.
• 6:08 - 6:12
And so the answer when we do
this subtraction 4 + 7 I take
• 6:12 - 6:17
away 2 + 3. I gives us a
complex number 2 + 4 I and
• 6:17 - 6:21
again we can see that what
we've done is. We know this,
• 6:21 - 6:26
I'm subtracting the real parts
4 takeaway two to give us two
• 6:26 - 6:29
and subtracted the imaginary
part 7 takeaway three to give
• 6:29 - 6:33
us four and that goes with the
eye because that's the
• 6:33 - 6:33
imaginary part.
• 6:35 - 6:40
The one last example now just to
make sure we've got this well
• 6:40 - 6:43
and truly understood, we're
• 6:43 - 6:45
complex number 5 + 6 I.
• 6:47 - 6:53
I'm going to take away from that
the complex number 7 - 3 I.
• 6:55 - 6:58
Let me just repeat the process.
We've seen several Times Now we
• 6:58 - 7:00
remove the brackets to start off
• 7:00 - 7:04
with. So nothing happens to
the first 2 terms, but the
• 7:04 - 7:07
minus operates on everything
that's in this next set of
• 7:07 - 7:09
brackets. So we get minus 7.
• 7:10 - 7:15
Then we get minus minus three.
I2 minus is making a plus.
• 7:16 - 7:21
Three, I and now we collect
together the real numbers we
• 7:21 - 7:23
collect together the imaginary
• 7:23 - 7:30
numbers. We have 5 - 7 which
is minus two. We have plus 6I
• 7:30 - 7:34
plus three I which is +9 I.
• 7:36 - 7:41
And so that's our answer. This
time and again we see that what
• 7:41 - 7:45
we've done is, we've subtracted
the real parts 5 - 7 gives us
• 7:45 - 7:49
minus two, and then we've
subtracted the imaginary parts 6
• 7:49 - 7:53
minus minus three is 6 + 3,
which is 9 because there's the
• 7:53 - 7:56
imaginary parts. That's nine I
• 7:57 - 8:02
So the way we add or subtract
2 complex numbers is simply to
• 8:02 - 8:06
use algebraic manipulation and
collect together like terms in
• 8:06 - 8:10
the next unit. We look at how
to multiply 2 complex numbers
• 8:10 - 8:11
together.
Title: