Video Transcript
Simplify two minus two π divided by three minus π.
In order to divide two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. This is because when we multiply a complex number by its conjugate, the answer is a purely real number.
We recall that for any complex number π§, which is equal to π₯ plus π¦π, then the conjugate, written π§ bar, is equal to π₯ minus π¦π. In this question, the conjugate of three minus π is three plus π. We need to multiply the fraction two minus two π over three minus π by the fraction three plus π over three plus π. We can now expand the parentheses on the numerator and denominator using the FOIL method.
On the numerator, the first terms have a product of six. Multiplying the outer terms gives us two π. The inner terms gives us negative six π. And multiplying the last terms gives us negative two π squared. On the denominator, we get nine plus three π minus three π minus π squared. Plus three π minus three π is equal to zero. We recall from our knowledge of imaginary and complex numbers that π squared is equal to negative one. The numerator therefore simplifies to eight minus four π, as six plus two is equal to eight and two π minus six π is negative four π. On the denominator, we have nine plus one.
Our expression is simplified to eight minus four π divided by 10. All three of our integers are even, so we can divide each term by two. The expression simplifies to four minus two π divided by or over five. For any simplification of this type, we can leave our answer as shown or we can split the real and imaginary parts. Four minus two π over five is equivalent to four-fifths minus two-fifths π.
The real part of our complex number is four-fifths, and the imaginary part is negative two-fifths. These are the values of π₯ and π¦ in the complex number π§ is equal to π₯ plus π¦π, respectively.
