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.