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.