Video Transcript
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.
