www.mathcentre.ac.uk/.../07-DivisionF61Mb.mp4

In this unit, we're going to
look at how to divide 2 complex

numbers. Now, division of
complex numbers is rather more

complicated than addition,
Subtraction, and multiplication.

And Division of complex numbers
relies on two very important

principles. The first is that
when you take a complex number

and multiply by its complex
conjugate, you get a real

number. The second important
principle is that when you have

a fraction, you can multiply the
numerator and the denominator.

That's the number on the top on
the number on the bottom of the

fraction by the same value, and
not change the value of a

fraction. So for example, if
you start with a fraction of

half and you multiply the
top and bottom by 5, you get

5/10 and the value of five
10s is the same as the value

of 1/2. And that's really
going to be very important

when we come into being able
to workout. How to divide 1

complex number by another.
So let's look at an example.

So we're going to take the
complex #4 + 7 I. I'm going to

divide it by the complex number
1 - 3. I now remember the

division is the same thing. It's
a fraction, so this complex

number divided by this one. We
can just write a Swan complex

number over another complex

number. So now we have a
fraction we can say is that we

won't change the value of this
fraction if we multiply the

numerator and the denominator by
the same value.

I'm going to choose to multiply
the denominator by 1 + 3 I.

1 + 3 I is the complex conjugate
of 1 - 3 I and we choose this

complex conjugate so that when
we do the multiplication, what's

in the denominator will turn out
to be a real number.

So for multiplying the
denominator by 1 + 3 I we've got

to multiply the numerator by 1 +

3 I. So that way we have
multiplied the numerator and

denominator by the same value,
so we haven't changed the value

of the answer.

So let's now multiply these two
fractions together. We multiply

out the two terms in the
numerator. We multiply out the

two terms in the denominator, so
we get 4 * 1 is 4.

4 * 3 I is 12 I.

Seven 8 * 1 is 7 I.

+78 times plus three. I is plus
21 by squares.

So that's multiplied. The two
terms in the numerator. Now we

multiply the two terms in the
denominator to get 1 * 1 one

times plus 3I.

Minus three items one.

And minus three I times
plus three. I give this

minus nine I squared.

What time do this up?

21 I squared is 21 times minus
one, so that's minus 21, so

we've got 4 - 21 is minus 17.

12 + 7 I is 99, so we've
got plus 99.

And then in the denominator.

I squared is minus one, so we've
got minus nine times minus one

is plus nine, 1 + 9 is 10.

And three I minus three. I is
nothing. So the management turns

disappear. So we've ended up
with a real denominator so we

could leave our answer like
this. Or we could split it up as

minus 17 over 10 + 19 over 10

I. And if we want we
could write as minus one point 7

+ 10.9 I.

So that's our answer. When we
divide 4 + 7, I buy 1 - 3.

I get minus one point 7 + 1.9.

Now let's do another example to
illustrate the principals again.

Here are two more complex
numbers 2 - 5 I and minus 4 + 3

i's going to divide the first
one by the second one.

And we write those as a fraction
2 - 5 I over minus 4 +

3. I now the way to do it
is to multiply. Want to multiply

the denominator by its complex
conjugate, which is minus 4 - 3

I. And because we're multiplying
the denominator by this value,

we must multiply the numerator.

By this value as well.

Now we multiply out the
numerator and denominator.

So we have two times minus four
is minus 8 two times minus

three. I is minus six I.

Minus 5I Times minus four is
plus 20I and minus 5I times

minus three. I is plus 15
I squared.

And then in the dominator we
have minus four times minus 4

inches 16. Minus four
times minus three I,

which is plus 12 I.

Plus three I times minus 4
inches minus 12 I.

I'm plus three I times
minus three I, which is

minus nine. I squared.

And now he tidies up.

59 squared is minus 15, so we've
got minus 8 - 15 is minus 23.

Minus six I plus
20I is plus 49.

So that's the numerator
simplified, and then the

denominator. We've got minus
nine. I squared, so that's plus

nine. We got 16 + 9 is
25 and 12. I minus 12. I

that disappears, leaving us with
a real denominator, which is

what we wanted. So we can write
that as minus 23 over 25 +

14 over 25 I.

Which we could also write
us minus N .92 +

.56 high.

And so that's the result
of doing this division.

Now in the next unit, we'll
look at something called the

organ diagram, which is a way
of graphically representing

complex numbers.