WEBVTT

00:00:02.990 --> 00:00:06.721
In this unit, we're going to
look at how to add and subtract

00:00:06.721 --> 00:00:11.425
complex numbers. Now when you're
at school, you first learn to

00:00:11.425 --> 00:00:14.710
add and subtract using the
counting numbers. That's numbers

00:00:14.710 --> 00:00:16.170
1234 and so on.

00:00:17.020 --> 00:00:20.814
And every time you meet a new
set of numbers, you learn a new

00:00:20.814 --> 00:00:24.337
rule or a new process for how
you can add and subtract them.

00:00:24.337 --> 00:00:27.860
So for example, when you meet
fractions, you learn it to add 2

00:00:27.860 --> 00:00:30.570
fractions, you must write them
both over a common denominator.

00:00:31.450 --> 00:00:35.614
No, it's the same with complex
numbers. You just need to learn

00:00:35.614 --> 00:00:39.778
the correct process to how to
add and subtract them, and with

00:00:39.778 --> 00:00:42.554
complex numbers is quite
straightforward. It's based on

00:00:42.554 --> 00:00:46.024
the principle you have seen
before in algebra where you

00:00:46.024 --> 00:00:49.841
collect together some like
terms, so we're going to do is

00:00:49.841 --> 00:00:53.311
going to start by adding two
algebraic expressions and then

00:00:53.311 --> 00:00:56.781
see how that carries over into
adding together 2 complex

00:00:56.781 --> 00:00:59.557
numbers. So let's take two
algebraic expressions. Let's

00:00:59.557 --> 00:01:01.292
take the expressions for plus.

00:01:01.470 --> 00:01:07.540
70 and 2
+ 3 two.

00:01:08.430 --> 00:01:13.435
So we have two expressions are
going to do is going to add

00:01:13.435 --> 00:01:18.055
these two things together, so
we're going to have 4 + 70.

00:01:19.110 --> 00:01:22.630
Plus 2 + 3 T.

00:01:24.390 --> 00:01:27.846
Now, the principle of collecting
together, like turns, simply

00:01:27.846 --> 00:01:31.302
says looking your long
expression and look for terms

00:01:31.302 --> 00:01:36.294
that are somehow the same. So in
this expression we have the four

00:01:36.294 --> 00:01:41.286
on the two, which is just
numbers. So we can put those two

00:01:41.286 --> 00:01:44.358
together. So 4 + 2 gives us 6.

00:01:45.340 --> 00:01:49.960
And we have these two terms
which have both got Tees in

00:01:49.960 --> 00:01:54.965
them, so we've got plus 70 + 3
more Tees so that gives

00:01:54.965 --> 00:01:56.505
altogether plus 10 T.

00:01:58.200 --> 00:02:02.640
So we started with two algebraic
expressions. We've added them

00:02:02.640 --> 00:02:07.132
up. And we simplified by
collecting together terms that

00:02:07.132 --> 00:02:12.254
are the same terms that are just
numbers and terms that have got

00:02:12.254 --> 00:02:15.406
teasing them. Now adding
together complex numbers works

00:02:15.406 --> 00:02:21.316
in exactly the same sort of way,
so we're going to do now is take

00:02:21.316 --> 00:02:26.044
two complex numbers and add them
together. So I'm going to take

00:02:26.044 --> 00:02:28.408
the complex #4 + 7 I.

00:02:29.360 --> 00:02:33.248
And the complex number 2 + 3 I.

00:02:34.120 --> 00:02:38.564
So this is first complex
number and this is my second

00:02:38.564 --> 00:02:41.796
complex number. I'm going to
add them together.

00:02:42.870 --> 00:02:44.854
Now what I do is I say well.

00:02:45.660 --> 00:02:50.364
If if I take away the brackets,
I won't have changed anything.

00:02:56.300 --> 00:02:59.600
Now I can look at this
expiration, which doesn't have

00:02:59.600 --> 00:03:03.890
any brackets in and I can look
for terms that I can collect

00:03:03.890 --> 00:03:08.180
together so I can collect the
four in the two together to get

00:03:08.180 --> 00:03:12.800
6, and I can collect the plus
Seven I and the plus three I

00:03:12.800 --> 00:03:14.450
together to get plus 10I.

00:03:17.240 --> 00:03:21.368
And so that's the answer. When I
add these two complex numbers

00:03:21.368 --> 00:03:26.184
together, I get this new complex
number 6 + 10 I and you'll see

00:03:26.184 --> 00:03:30.312
that in fact all we've done is
we've added together the real

00:03:30.312 --> 00:03:33.408
parts of the two complex
numbers, and we've added

00:03:33.408 --> 00:03:37.192
together the imaginary parts of
the two complex numbers to get

00:03:37.192 --> 00:03:41.258
the answer. So let's do
another example. OK, in this

00:03:41.258 --> 00:03:44.057
example will just take two
different complex numbers and

00:03:44.057 --> 00:03:44.990
add them together.

00:03:46.950 --> 00:03:50.480
So we'll take the complex
numbers 5 + 6 I.

00:03:51.650 --> 00:03:54.778
And the complex number 7 - 3 I.

00:03:56.340 --> 00:03:58.475
Will add those two complex

00:03:58.475 --> 00:04:04.016
numbers together. So just as
before will write everything out

00:04:04.016 --> 00:04:05.432
without any brackets.

00:04:08.960 --> 00:04:13.003
And then we look for the terms
that we can collect together so

00:04:13.003 --> 00:04:16.424
we can collect together the real
parts. That's just the numbers.

00:04:16.424 --> 00:04:18.912
The real numbers 5 + 7 to give

00:04:18.912 --> 00:04:24.350
us 12. And then we can collect
together the two imaginary parts

00:04:24.350 --> 00:04:28.450
plus six I minus three I giving
us plus 3I.

00:04:31.220 --> 00:04:34.828
And so the answer when we had
these two complex numbers

00:04:34.828 --> 00:04:39.748
together is 12 + 3 I and once
again we see that we've done is.

00:04:39.748 --> 00:04:43.028
We've just added together the
real parts and we've added

00:04:43.028 --> 00:04:44.340
together the imaginary parts.

00:04:45.980 --> 00:04:48.950
Now subtraction works in
exactly the same way, we just

00:04:48.950 --> 00:04:52.811
have to be careful that we
make sure that we do take away

00:04:52.811 --> 00:04:55.187
the whole of the second
complex number. 'cause

00:04:55.187 --> 00:04:58.157
sometimes people might just
take away the real part and

00:04:58.157 --> 00:05:01.127
forget to take away the
imaginary part as well. So

00:05:01.127 --> 00:05:03.206
let's see how we can do that.

00:05:04.370 --> 00:05:10.430
So if we go back to our first
pair of numbers 4 + 7 I.

00:05:11.220 --> 00:05:15.730
We're going to take away from
that 2 + 3 I.

00:05:18.310 --> 00:05:22.236
So when we remove the brackets,
this time we have to be careful

00:05:22.236 --> 00:05:25.860
to make the minus operate on the
whole of this complex number.

00:05:26.620 --> 00:05:30.700
So from the first complex number
we have 4 + 7 I.

00:05:32.110 --> 00:05:35.818
But then when we take these
brackets out, we get minus 2.

00:05:37.990 --> 00:05:42.202
Then we get minus plus three
eyes are getting minus three I.

00:05:43.590 --> 00:05:46.505
And now we're in a position to
collect together like terms.

00:05:46.505 --> 00:05:49.420
Just as we've been doing in all
the examples so far.

00:05:50.950 --> 00:05:54.734
So our numbers are 4 - 2 which

00:05:54.734 --> 00:06:00.370
is 2. And then our terms
with I in them are plus

00:06:00.370 --> 00:06:04.660
Seven. I minus three I,
so that gives us plus 4I.

00:06:07.590 --> 00:06:12.336
And so the answer when we do
this subtraction 4 + 7 I take

00:06:12.336 --> 00:06:17.421
away 2 + 3. I gives us a
complex number 2 + 4 I and

00:06:17.421 --> 00:06:21.489
again we can see that what
we've done is. We know this,

00:06:21.489 --> 00:06:25.557
I'm subtracting the real parts
4 takeaway two to give us two

00:06:25.557 --> 00:06:28.947
and subtracted the imaginary
part 7 takeaway three to give

00:06:28.947 --> 00:06:32.676
us four and that goes with the
eye because that's the

00:06:32.676 --> 00:06:33.354
imaginary part.

00:06:35.400 --> 00:06:39.820
The one last example now just to
make sure we've got this well

00:06:39.820 --> 00:06:43.220
and truly understood, we're
going to take start with a

00:06:43.220 --> 00:06:45.260
complex number 5 + 6 I.

00:06:46.960 --> 00:06:53.106
I'm going to take away from that
the complex number 7 - 3 I.

00:06:54.580 --> 00:06:58.228
Let me just repeat the process.
We've seen several Times Now we

00:06:58.228 --> 00:07:00.052
remove the brackets to start off

00:07:00.052 --> 00:07:04.270
with. So nothing happens to
the first 2 terms, but the

00:07:04.270 --> 00:07:07.370
minus operates on everything
that's in this next set of

00:07:07.370 --> 00:07:09.230
brackets. So we get minus 7.

00:07:10.390 --> 00:07:14.926
Then we get minus minus three.
I2 minus is making a plus.

00:07:16.260 --> 00:07:21.221
Three, I and now we collect
together the real numbers we

00:07:21.221 --> 00:07:23.025
collect together the imaginary

00:07:23.025 --> 00:07:30.453
numbers. We have 5 - 7 which
is minus two. We have plus 6I

00:07:30.453 --> 00:07:33.890
plus three I which is +9 I.

00:07:36.220 --> 00:07:40.731
And so that's our answer. This
time and again we see that what

00:07:40.731 --> 00:07:45.242
we've done is, we've subtracted
the real parts 5 - 7 gives us

00:07:45.242 --> 00:07:48.712
minus two, and then we've
subtracted the imaginary parts 6

00:07:48.712 --> 00:07:53.223
minus minus three is 6 + 3,
which is 9 because there's the

00:07:53.223 --> 00:07:55.999
imaginary parts. That's nine I
in the answer.

00:07:57.090 --> 00:08:02.173
So the way we add or subtract
2 complex numbers is simply to

00:08:02.173 --> 00:08:05.692
use algebraic manipulation and
collect together like terms in

00:08:05.692 --> 00:08:10.384
the next unit. We look at how
to multiply 2 complex numbers

00:08:10.384 --> 00:08:10.775
together.