Video Transcript
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.
