Video Transcript
In this video, we’ll learn how to
divide complex numbers. We’ll learn first how to divide a
complex number by a real and then a purely imaginary number, before generalizing
these techniques to allow us to divide a complex number by another complex
number. We’ll then learn how to use these
processes to solve equations involving the division of complex numbers.
Let’s begin by learning how to
divide a complex number by a real number. This is very much an extension of
the process we use to multiply complex numbers by real numbers. We multiply a complex number by a
real number by using the distributive property and multiplying each part by the real
number. We can think of division by a real
number 𝑐 as the same as multiplying by the reciprocal of 𝑐, multiplying by one
over 𝑐. One over 𝑐 multiplied by 𝑧 is 𝑧
divided by 𝑐. We multiply one over 𝑐 by 𝑎 and
we get 𝑎 over 𝑐. And we multiply one over 𝑐 by 𝑏𝑖
and we get 𝑏 over 𝑐𝑖. We can see then that, to divide a
complex number by a real number 𝑐, we simply divide the real part and then divide
the imaginary part.
Let’s look at an example.
Given that 𝑧 is equal to five
plus three 𝑖, express 𝑧 over two in the form 𝑎 plus 𝑏𝑖.
We’ve been given a complex
number, five plus three 𝑖, and we’re looking to work out the value or the
complex number given by 𝑧 divided by two. 𝑧 divided by two is five plus
three 𝑖 divided by two. To divide a complex number by a
real number, we need to divide both the real and the imaginary part of the
complex number by that real number. We divide five — that’s the
real part — by two. And we divide the imaginary
part, three, by two. So we see that, for our complex
number, 𝑧 over two is the same as five over two plus three over two 𝑖.
And what about dividing a complex
number by a purely imaginary number?
Simplify two plus four 𝑖 over
𝑖.
To work out how to divide two
plus four 𝑖 by 𝑖, we recall the definition of 𝑖. It’s a solution to the equation
𝑥 squared equals negative one. And we say that 𝑖 squared is
equal to negative one. Or often 𝑖 is equal to the
square root of negative one.
If we consider this fraction as
two plus four 𝑖 divided by the square root of negative one, we can see that, to
simplify, we’d need to perform the same process as rationalizing the denominator
when we’re dealing with any other radical. We multiply both the numerator
and the denominator of our fraction by the square root of negative one.
In fact, we know that the
square root of negative one is 𝑖. So we’re going to be
multiplying both the numerator and the denominator of this fraction by 𝑖. And we’re allowed to do that
because multiplying by 𝑖 over 𝑖 is the same as multiplying by one. Essentially, we’re creating an
equivalent fraction.
Let’s apply the distributive
property for 𝑖 multiplied by two plus four 𝑖. 𝑖 multiplied by two is two 𝑖,
and 𝑖 multiplied by four 𝑖 is four 𝑖 squared. Now of course 𝑖 squared is
equal to negative one. So our expression becomes two
𝑖 plus four multiplied by negative one, which is negative four plus two 𝑖.
On the denominator, we have 𝑖
multiplied by 𝑖, which is of course 𝑖 squared, which is negative one. So we can rewrite two plus four
𝑖 over 𝑖 as negative four plus two 𝑖 over negative one. And now we’re simply dividing
by a real number. And to divide a complex number
by a real number, we divide the real part and then separately divide the
imaginary part. Negative four divided by
negative one is four, and two 𝑖 divided by negative one is negative two 𝑖. And we fully simplify two plus
four 𝑖 over 𝑖. It’s four minus two 𝑖.
In fact, we can use a similar
technique to help us divide two complex numbers. Just as we’ve used the rules for
rationalizing the denominator when that denominator is a radical, we can also apply
the rules for rationalizing a denominator when that denominator is an expression
involving a rational number and a radical.
Let’s consider an example.
Simplify three minus six 𝑖
over one minus five 𝑖.
To simplify this fraction or
divide three minus six 𝑖 by one minus five 𝑖, we need to find a way to get a
real number as our denominator. So what can we do to get that
real number? Well, remember, if we multiply
a complex number by its complex conjugate — that’s found by changing the sign of
the imaginary part — we end up with a real number. So for a complex number of the
form 𝑎 plus 𝑏𝑖, its conjugate is 𝑎 minus 𝑏𝑖.
So if we multiply the numerator
and denominator of this fraction by the complex conjugate of one minus five 𝑖,
we’ll end up with a real denominator. And the conjugate of one minus
five 𝑖 is one plus five 𝑖. We then distribute these
parentheses as normal. There’s a number of different
methods we could use.
Let’s look at the FOIL
method. “F” stands for “first.” We multiply the first term in
the first bracket by the first term in the second bracket. That’s three. “O” stands for “outer.” We multiply the outer
terms. That’s 15𝑖. “I” stands for “inner.” We multiply the inner
terms. That’s negative six 𝑖. And “L” stands for “last.” We multiply the last term in
each bracket. And this time, we get negative
30𝑖 squared.
Since we know that 𝑖 squared
is equal to negative one, our last term becomes negative 30 multiplied by
negative one, which is 30. And we can simplify our
numerator to 33 plus nine 𝑖.
And we’ll repeat this process
with the denominator. We get one plus five 𝑖 minus
five 𝑖 minus 25𝑖 squared. And of course, 𝑖 squared is
negative one. So we end up with our final
term being positive 25. And then five 𝑖 minus five 𝑖
is zero. So we’re left with 26. And we have that real
denominator we were looking for. So we’ve partly simplified our
fraction. We now have 33 plus nine 𝑖
divided by 26.
To divide a complex number by a
real number though, we can divide the real part and then separately divide the
imaginary part. The real part of our answer is
33 over 26, and the imaginary part is nine over 26. So our fraction in its simplest
form is 33 over 26 plus nine over 26𝑖.
And in fact, we could’ve saved
ourselves a little bit of time by recalling the general formula for the product
of a complex number with its conjugate. If the complex number is of the
form 𝑎 plus 𝑏𝑖, we say that that can be found by 𝑎 squared plus 𝑏
squared. So in general, to divide a
complex number by another complex number, we write it as a fraction. And we then multiply both the
numerator and the denominator of this fraction by the conjugate of the
denominator. We can then distribute and
simplify as far as possible.
Let’s see what this looks like in
general.
One: Expand and simplify 𝑝
plus 𝑞𝑖 multiplied by 𝑝 minus 𝑞𝑖. Two: Expand 𝑎 plus 𝑏𝑖
multiplied by 𝑝 minus 𝑞𝑖. Three: Hence, find a fraction
which is equivalent to 𝑎 plus 𝑏𝑖 over 𝑝 plus 𝑞𝑖 and whose denominator is
real.
For part one of this question,
we’re looking to multiply two complex numbers. We could absolutely apply any
method for distributing brackets, such as the FOIL method or the grid
method. However, if we look really
carefully, we can see that these two complex numbers are conjugates of one
another.
For a complex number of the
form 𝑎 plus 𝑏𝑖, where 𝑎 is the real part and 𝑏 is the imaginary part, its
conjugate is found by changing the sign of the imaginary part. And this is really useful. It allows us to use a formula
for the product of a complex number with its conjugate. It’s 𝑎 squared plus 𝑏
squared. We square the real part and add
it to the square of the imaginary part. The real part of our complex
number is 𝑝, and the imaginary part is 𝑞. So the product of 𝑝 plus 𝑞𝑖
with its conjugate 𝑝 minus 𝑞𝑖 is 𝑝 squared plus 𝑞 squared.
Unfortunately, there are no
nice tricks that will allow us to multiply 𝑎 plus 𝑏𝑖 with 𝑝 minus 𝑞𝑖. We’ll use the FOIL method
instead. We multiply the first
terms. 𝑎 multiplied by 𝑝 is
𝑎𝑝. We multiply the outer terms,
and we get 𝑎𝑞𝑖. Multiplying the inner terms
gives us 𝑏𝑝𝑖. And clearing a little space to
multiply the last terms, we get negative 𝑏𝑞𝑖 squared.
Now in fact, we should recall
that 𝑖 squared is equal to negative one. And so this last term becomes
positive 𝑏𝑞. We’re going to rearrange this a
little so it looks like a complex number. We add the real parts and we
get 𝑎𝑝 plus 𝑏𝑞. And we separately add the
imaginary parts. And when we do, we find that
the imaginary part of the distribution of these brackets is 𝑏𝑝 minus 𝑎𝑞. So the answer to part two is
𝑎𝑝 plus 𝑏𝑞 plus 𝑏𝑝 minus 𝑎𝑞𝑖.
And the final part is to find
an equivalent fraction to 𝑎 plus 𝑏𝑖 over 𝑝 plus 𝑞𝑖. And of course, it’s no
coincidence that we’ve been asked to do the working out that we already
have. We want to create an equivalent
fraction which has a real denominator. To achieve this, we multiply
both the numerator and the denominator of our fraction by the complex conjugate
of the denominator. And of course, we already
evaluated these.
So we see that an equivalent
fraction to 𝑎 plus 𝑏𝑖 over 𝑝 plus 𝑞𝑖 whose denominator is real — and in
fact the general form of 𝑎 plus 𝑏𝑖 divided by another complex number 𝑝 plus
𝑞𝑖 — is 𝑎𝑝 plus 𝑏𝑞 plus 𝑏𝑝 minus 𝑎𝑞𝑖 all over 𝑝 squared plus 𝑞
squared. Now remember, whilst it’s all
fine and well to derive this formula, it’s important to focus on applying the
processes each time.
If 𝑎 plus 𝑏𝑖 equals negative
three minus five 𝑖 over negative three plus five 𝑖, is it true that 𝑎 squared
plus 𝑏 squared equals one?
We’ve been given the quotient
of two complex numbers. And we’re told that this can be
expressed as a single complex number 𝑎 plus 𝑏𝑖. In order to evaluate the
expression 𝑎 squared plus 𝑏 squared, we’re going to need to work out what this
single complex number actually is. To do so, we apply the
processes for dividing complex numbers.
We need to multiply both the
numerator and denominator of this fraction by the conjugate of the
denominator. We find the conjugate by
changing the sign of the imaginary part. And if we do that, we see that
the conjugate of our denominator is negative three minus five 𝑖.
We’re then going to multiply by
applying the distributive property. Let’s begin with the
numerator. We’re going to use the FOIL
method. Negative three multiplied by
negative three is nine. Multiplying the outer terms, we
get 15𝑖. And indeed the inner terms
gives us 15𝑖. And when we multiply the last
two terms, we get 25𝑖 squared. But of course, 𝑖 squared is
equal to negative one. So this last term is 25
multiplied by negative one, which is negative 25. We then add the real parts and
separately add the imaginary parts, although we can think about this like
collecting like terms. And we get the numerator of our
fraction to be negative 16 plus 30𝑖.
Now we could repeat this
process for the denominator. However, if we recall, for a
general complex number of the form 𝑎 plus 𝑏𝑖 whose conjugate is of course 𝑎
minus 𝑏𝑖, the product of these two numbers is found by 𝑎 squared plus 𝑏
squared. The real part of our complex
number is negative three, and the imaginary part is five. So when we distribute these
parentheses, we’re going to get negative three squared plus five squared. That’s nine plus 25, which is
34. So negative three minus five 𝑖
over negative three plus five 𝑖 is negative 16 plus 30𝑖 over 34.
And of course, if we’re
dividing a complex number by a real number, we divide the real parts and then
the imaginary parts separately. Negative 16 over 34 simplifies
to negative eight over 17. And 30 over 34 simplifies to 15
over 17. And so we can see that 𝑎 must
be equal to negative eight over 17 and 𝑏 must be equal to 15 over 17.
All that’s left is to consider
the sum of their squares, 𝑎 squared plus 𝑏 squared. That’s negative eight over 17
squared plus 15 over 17 squared. They’ll have the same
denominator. So that’s 64 plus 225 over
289. But actually, 64 plus 225 is
289. So 𝑎 squared plus 𝑏 squared
equals 289 over 289, which is of course one. And it is indeed true that 𝑎
squared plus 𝑏 squared equals one.
Now in fact, in this example,
the two complex numbers we were dividing were conjugates of one another. It is actually a general rule
that if we divide a complex number by its conjugate, the sum of the squares of
the real and imaginary part of that new complex number will be one.
We’ll now consider how to solve
equations which involve the division of complex numbers.
Solve the equation 𝑧
multiplied by two plus 𝑖 equals three minus 𝑖 for 𝑧.
To solve this equation for 𝑧,
we’ll need to apply inverse operations. We’ll begin by dividing both
sides of this equation by two plus 𝑖. And we see that 𝑧 is equal to
three minus 𝑖 divided by two plus 𝑖. To divide three minus 𝑖 by two
plus 𝑖, we’re going to need to multiply both the numerator and the denominator
of the fraction by the conjugate of two plus 𝑖. To find the conjugate, we
change the sign of the imaginary part. And we see that the conjugate
of two plus 𝑖 is two minus 𝑖.
We’ll distribute the
parentheses at the top of this fraction by using the FOIL method. Three multiplied by two is
six. Three multiplied by negative 𝑖
is negative three 𝑖. We then get negative two
𝑖. And our last term gives us 𝑖
squared. 𝑖 squared is of course
negative one. So our last term is negative
one.
And we can collect like terms
or add the real parts and separately add the imaginary parts. And we see that three minus 𝑖
multiplied by two minus 𝑖 is five minus five 𝑖. And we could repeat this
process for the denominator.
However, there is a special
rule we can use to multiply a complex number by its conjugate. We can find the sum of the
squares of the real and imaginary parts. The real part is two, and the
imaginary part, the coefficient of 𝑖, is one. So the product of these two
complex numbers is four plus one, which is five. And we see that 𝑧 is equal to
five minus five 𝑖 over five.
We can then divide the real
parts by this real number. We get five divided by five,
which is one. And separately we divide the
imaginary part by this real number. Five divided by five is
one. So we get one minus 𝑖. And we’ve solved our equation
for 𝑧.
We’ve seen that division of complex
numbers can be quite time-consuming.
We’ll look at one final example of
where we can simplify our working somewhat.
Simplify three minus four 𝑖
over two plus two 𝑖 plus three minus four 𝑖 over two minus two 𝑖.
In this question, we’re looking
to find the sum of two fractions whose denominators and numerators are both
complex numbers. We could apply the rules for
dividing complex numbers and work from there. However, that’s quite a lengthy
process, especially for two fractions. Instead, we notice that the
numerator of each fraction is the same. And we can therefore rewrite
this expression by taking out a factor of three minus four 𝑖. And we have three minus four 𝑖
multiplied by one over two plus two 𝑖 plus one over two minus two 𝑖.
Next, we’ll add these fractions
by finding a common denominator. The common denominator is the
product of these two numbers. It’s two plus two 𝑖 multiplied
by two minus two 𝑖. And when we multiply the
numerator of the first fraction by two minus two 𝑖, we get two minus two
𝑖. And for the numerator of the
second fraction, we get two plus two 𝑖. So we’ll simplify this
next.
For the numerator, negative two
𝑖 plus two 𝑖 is zero. So we’re simply left with
four. And we won’t actually expand
the brackets on the denominator. Instead, we use the fact that
they are complex conjugates of one another. And we can find their product
by finding the sum of the squares of the real parts and the imaginary parts. That’s two squared plus two
squared, which is eight.
Now four over eight simplifies
to one-half. So we need to find one-half of
three minus four 𝑖. A half of the real part is
three over two, and a half of the imaginary part is negative two. So our solution is three over
two minus two 𝑖.
In this video, we’ve learned that
we can divide complex numbers by using the same techniques as we do when
rationalizing the denominator. And that’s by multiplying both the
numerator and the denominator of the fraction by the conjugate of the denominator,
expanding the brackets, and then simplifying. We’ve also seen that it can be
useful to look for common factors to help simplify any more complicated
expressions.
